L L L L L L L
a a a a a a a a
1 2 3 4 5 6 7 8
46
L
Thususingasingleordinarycolumnmatrixofrefined labelsweobtainmanysupercolumnvectorsofrefinedlabels. SowecanforeachtypeofpartitiononVgetagroupunder addition.Thiswillbeexhibitedfromthefollowing: Consider X=
a a a a
1 2 3 4
L L L L beacolumnvectorofrefinedlabels. X1=
L L L L
a a a a
1 2 3 4
,X2=
L L L L
a a a a
1 2 3 4
,X3=
L L L L
a a a a
1 2 3 4
,X4=
L L L L
a a a a
1 2 3 4
, X5=
L L L L
a a a a
1 2 3 4
,X6=
L L L L
a a a a
1 2 3 4
andX7=
L L L L
a a a a
1 2 3 4
.
Thuswehavesevensupercolumnvectorsofrefinedlabels foragivencolumnvectorrefinedlabelX. Nowif T 1=
L L XLL;1i4 L L
a a 1aR a a
=∈≤≤
1 2 i 3 4
bethecollectionofallsupercolumnvectorsofrefinedlabels. ThenT 1isagroupunderadditionofsupercolumnvectorsof refinedlabels.
47
Likewise T 2=
L L XLL;1i4 L L
1 2 i 3 4
a a 2aR a a
=∈≤≤ isasupercolumnvectorgroupofrefinedlabels.
ThusT i= {} i iaR XLL;1i4 ∈≤≤ isasupercolumnvector ofrefinedlabelsi=1,2,3,…,7.Hencewegetsevendistinct groupsofsupercolumnvectorsofrefinedlabels.
ConsiderY= () 12345 aaaaa LLLLLbeasuperrows vector(matrix)ofrefinedlabels. Y1= () 12345 aaaaa LLLLL,Y2= () 12345 aaaaa LLLLL Y3= () 12345 aaaaa LLLLL,Y4= () 12345 aaaaa LLLLL, Y5= () 12345 aaaaa LLLLL,Y6= () 12345 aaaaa LLLLL, Y7= () 12345 aaaaa LLLLL,Y8= () 12345 aaaaa LLLLL, Y9= () 12345 aaaaa LLLLL,Y10= () 12345 aaaaa LLLLL, Y11= () 12345 aaaaa LLLLL,Y12= () 12345 aaaaa LLLLL, Y13= () 12345 aaaaa LLLLL,Y14= () 12345 aaaaa LLLLL andY15= () 12345 aaaaa LLLLL.
Thuswehave15superrowvectorsofrefinedlabelsbuilt usingtherowvectorYofrefinedlabels.Hencewehave15 groupsassociatedwiththesinglerowvector
X= () 12345 aaaaa LLLLL ofrefinedlabels. Thisistheadvantageofusingsuperrowvectorsintheplace ofrowvectorsofrefinedlabels. Considerthematrix
48
P=
aaa aaa aaa aaa
123 456 789 101112
LLL LLL LLL LLL ofrefinedlabels.Howmanysupermatricesofrefinedlabelscan begotfromP. P1=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
,P2=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
, P3=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
,P4=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
, P5=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
,P6=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
, P7=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
,P8=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
, P9=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
,P10=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
,
49
P11=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
P12=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
P13=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
,P14=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
,
,
P15=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
,P16=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
, P17=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
,P18=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
, P19=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
,P20=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
, P21=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
,P22=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
,
50
LLL LLL LLL LLL
aaa aaa aaa aaa
,P24=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
, P25=
LLL LLL LLL LLL
aaa aaa aaa aaa
123 456 789 101112
,P26=
LLL LLL LLL LLL and P27=
aaa aaa aaa aaa
123 456 789 101112
123 456 789 101112
. ThusthematrixPofrefinedlabelsgivesus27supermatrix ofrefinedlabelstheirbywecanhave27groupsofsuper matricesofrefinedlabelsusingP 1,P2,…,P27.Nowsupposewe studythesupermatrixofrefinedlabelsusingR{0} L +
∪ ={Lr|r
aaa aaa aaa aaa ∈ R+ ∪ {0}}thenalsowecangetthesupermatrixofrefined
labels.Inthiscaseanysupermatrixofrefinedlabelscollection ofaparticulartypemaynotbeagroupbutonlyasemigroup underaddition.Let
LLL LLL LLL LLL X= () {} 123456i aaaaaaaR{0} LLLLLLLL + ∪ ∈ beacollectionofsuperrowvectorofrefinedlabelswithentries from R{0} L + ∪ .ClearlyXisonlyasemigroupasnoelementhas inverse. Thuswehavethefollowingtheorem. THEOREM 2.3: Let X = () {} + ∪ ∈≤≤ 123n1ni aaaaaa R{0} LLL...LLLL;1in
51
P23=
123 456 789 101112
be a collection of super row vectors of refined labels. X is a commutative semigroup with respect to addition.
Theproofisdirectandsimpleandhenceleftasanexerciseto thereader.
bethesupercolumnvectorofrefinedlabels.
vectorsofrefinedlabelsxandyfromR{0}
areequivalentor similarorofsametypeifbothxandyhavethesamenatural orderandenjoyidenticalpartitiononit.
labelsonlywhentheyhavesimilarpartition.
52
Nowconsider M= 1 2 3 4 i 5 6 n a a a a aR{0} a a a L L L L LLL;1in L L + ∪ ∈≤≤
Nowwesayasincaseofotherlabelsthattwosupercolumn
L + ∪
Wecanaddonlythosetwosupercolumnvectorrefined
Consider X= 1 2 3 4 5 6 7 a a a a a a a L L L L L L L andY= 1 2 3 4 5 6 7 b b b b b b b L L L L L L L where i a L ∈ R{0} L + ∪ ;1 ≤ i ≤ 7.
X+YisdefinedandX+Yisagainasupercolumnvectorof refinedlabelsofsametype.Inviewofthiswehavethe followingtheorem.
THEOREM 2.4: Let M = + ∪ ∈≤≤
1 2 3 i n2 n1 n
a a a a R{0} a a a
LL;1in
be the collection of super column vectors of refined labels of same type with entries from {0} R L + ∪ . M is a semigroup under addition.
Thisproofisalsodirectandhenceleftasanexercisetothe reader.
labelsareadded.
53
L L L
L L L
L L L L
Nowweshowhowtwosupercolumnvectorsofrefined
Let X= 1 2 3 4 5 6 7 8 a a a a a a a a
L L L L andY= 1 2 3 4 5 6 7 8 b b b b b b b b L L L L L L L L
54 beanytwosupercolumnvectorsofrefinedlabels.BothXand
refinedlabelswehave X= 123 456 789 101112 131415 161718 aaa aaa aaa aaa aaa aaa LLL LLL LLL LLL LLL LLL isasupercolumnvectorofrefinedlabelsfromthesetR{0} L + ∪ . Likewise Y= 159131721 2610141822 i 3711151923 4812162024 aaaaaa aaaaaa aR{0} aaaaaa aaaaaa LLLLLL LLLLLL LL LLLLLL LLLLLL + ∪ ∈ isasuperrowvectorofrefinedlabelsanditiseasilyverifiedY isnotasimplesuperrowvectorofrefinedlabels. NowitisinterestingtonoteR{0} L + ∪ ={setofallrefined labelswithpositivevalue}isasemifieldofrefinedlabelsor DSmsemifieldofrefinedlabelsand R{0} L + ∪ ≅ R+ ∪ {0}[48]. Thusweseefurther A= 1234 5678 9101112 aaaa aaaa aaaa LLLL LLLL LLLL
YareofsametypeandX+YisalsoofthesametypeasXand Y.Weseenowasincaseofusualsupercolumnvectorof
55
isasupermatrixofrefinedlabelswithentriesfromR{0} L + ∪ . Suppose A= 1234 5678 9101112 13141516 aaaa aaaa aaaa aaaa LLLL LLLL LLLL LLLL beasupermatrixofrefinedlabelswithentriesfromR{0} L + ∪ . Nowifweconsidersupermatricesofrefinedlabelswithentries from R{0} L + ∪ ofsamenaturalorderandsimilartypeofpartition thenadditioncanbedefined. Wewilljustshowhowadditionisdefinedwhensuper matricesofrefinedlabelsareofsametype. LetA= 12345 678910 1112131415 1617181920 aaaaa aaaaa aaaaa aaaaa LLLLL LLLLL LLLLL LLLLL and B= 12345 678910 1112131415 1617181920 bbbbb bbbbb bbbbb bbbbb LLLLL LLLLL LLLLL LLLLL beanytwosupermatricesofrefinedlabelswithentriesfrom R{0} L + ∪ ;1 ≤ i ≤ 20. A+B= 12345 678910 1112131415 1617181920 aaaaa aaaaa aaaaa aaaaa LLLLL LLLLL LLLLL LLLLL +
bbbbb bbbbb bbbbb bbbbb
12345 678910 1112131415 1617181920
+++++ +++++ = +++++ +++++ =
LLLLLLLLLL LLLLLLLLLL LLLLLLLLLL LLLLLLLLLL
ababababab ababababab ababababab ababababab
LLLLL LLLLL LLLLL LLLLL 1122334455 667788991010 11111212131314141515 16161717181819192020
LLLLL LLLLL LLLLL LLLLL
ababababab ababababab ababababab ababababab
1122334455 667788991010 11111212131314141515 16161717181819192020
+++++ +++++ +++++ +++++
.
WeseeA+B,AandBareofsametype.Inviewofthiswe havethefollowingtheorem.
THEOREM 2.5: Let A = 123 456 789
aaa aaa aaa ai are submatrices of refined labels with entries from + ∪ R{0} L ; 1 ≤ i ≤ 9}. A is a semigroup under addition.
HereAisacollectionofallsupermatricesofrefinedlabels wherea 1,a 2anda 3aresubmatricesofrefinedlabelswithsame numberofrows.Alsoa 4,a 5anda 6aresubmatricesofrefined labelswithsamenumberofrows.a 7,a 8anda 9aresubmatrices ofrefinedlabelswithsamenumberofrows.
Similarlya 1,a 4anda 7aresubmatricesofrefinedlabelswith samenumberofcolumns.a 2,a 5anda 6aresubmatricesof
56
3,a 6anda 9are submatricesofrefinedlabelswithsamenumberofcolumns.
partitiononit.Wecallsuchsuperrowvectorsofordinarylabels assuperrowvectorsofsametype.
57
Let X= () 123456 aaaaaa LLLLLLand Y= () 123456 bbbbbb LLLLLL betwosuperrowvectorsofordinarylabels;ij abL,L ∈ L={0= L0<L1<L2<...<Lm<Lm+1=1},1 ≤ i,j ≤ 6.Now‘+’cannot bedefined.Wedefine‘∪’whichismax(ij ab L,L). ThusX ∪ Y= () 123456 aaaaaa LLLLLL ∪ () 123456 bbbbbb LLLLLL () 112233445566 abababababab LLLLLLLLLLLL =∪∪∪∪∪∪ =(max{11 ab L,L}|max{22 ab L,L},max{33 ab L,L}|max {44 ab L,L}max{55 ab L,L}max{66 ab L,L}) = () 123456 cccccc LLLLLL.
refinedlabelswithsamenumberofcolumns.a
ThusAisasemigroupunderaddition. Now Q{0} L + ∪ givesthecollectionofallrefinedlabelsand Q{0} L + ∪ ≅ Q+ ∪ {0},infactQ{0} L + ∪ ⊆ R{0} L + ∪ . Supposewetakethelabels0=L0<L1<L2<...<Lm= Lm+1=1,thatthecollectionofordinarylabels;thenwithminor maxorequivalently ∩ or ∪,theselabelsaresemilatticesor semigroups.ThatisL={0=L0<L1<L2<...<Lm<Lm+1=1} isasemilattice.Infactwhenboth ∪ and ∩ aredefinedLisa latticecalledthechainlattice. LetP= () {} 12345i aaaaaa LLLLLLL ∈ =L0<L1< ...<Lm<Lm+1=1}beasuperrowvectorofordinarylabels.We saytwosuperrowvectorordinarylabelsaresimilaroridentical iftheyareofsamenaturalorderandholdsameoridentical
WeseeX,YandX ∪ Yaresametypeofsuperrowvectors ofordinarylabels.
InananalogouswaywecandefineX ∩ Yormin{X,Y}. X ∩YisagainofthesametypeasXandY. Nowwehavethefollowinginterestingresults.
THEOREM 2.6: Let V = (){ 1234567n1ni aaaaaaaaaa LLLLLLL......LLL
∈ L= {0 = L0 < L1 < . . . < Lm < Lm+1= 1}; 1 ≤ i ≤ an} be the collection of super row vectors of ordinary labels. V is a semigroup under ‘∪’.
THEOREM 2.7: Let V= (){ 1234567n1ni aaaaaaaaaa LLLLLLL......LLL
∈ L = {0 = L0 < L1 < . . . < Lm = Lm+1 = 1}; 1 ≤ i ≤ an} be the collection of super row vectors of ordinary labels. V is a semigroup under ‘∩’.
Theproofisleftasanexercisetothereader.
Inthetheorem2.7if‘∩’isreplacedby‘∪’,Visa semigroupofsuperrowmatrices(vector)ofordinarylabels. Nowconsider P=
58
2 3 4 5 6 7 8
1
a a a a a a a a L L L L L L L L
samenaturalorder.Anytwosupercolumnvectorsofordinary labelsofsamenaturalordercanhave
onlywhenthebothofthemenjoythesametypeofpartitionon
59
betwosupercolumnvectormatricesofordinarylabelswhere ij abL,L ∈ {L;0=L0<L1<L2<...<L
∩
where i a L ∈ L={0=L0<L1 <...<Lm<Lm+1=1},1≤ i ≤ 8,Pisasupercolumnvectorof ordinarylabels.Nowwecannotingeneraldefine ∪ or ∩ onany twosupercolumnvectorsofordinarylabelseveniftheyareof
∪ or ∩ definedonthem
it. Wewillnowgiveafewillustrationsbeforeweproceedon tosuggestsomerelatedresults. Let T= 1 2 3 4 5 6 7 8 9 a a a a a a a a a L L L L L L L L L andS= 1 2 3 4 5 6 7 8 9 b b b b b b b b b L L L L L L L L L
m<Lm+1};1 ≤ i,j ≤ 9. Nowwedefinemin{T,S}=T
Sasfollowsmin{T,S}=
60
L
min{ 1 2 3 4 5 6 7 8 9 a a a a a a a a a L L L L L L L L L , 1 2 3 4 5 6 7 8 9 b b b b b b b b b L L L L L L L L L }= 11 22 33 44 55 66 77 88 99 ab ab ab ab ab ab ab ab ab min{L,L} min{L,L} min{L,L} min{L,L} min{L,L} min{L,L} min{L,L} min{L,L} min{L,L} =T ∩ S= 1 2 3 4 5 6 7 8 9 a a a a a a a a a L L L L L L L L L ∩ 1 2 3 4 5 6 7 8 9 b b b b b b b b b
L L L L L L L L = 11 22 33 44 55 66 77 88 99 ab ab ab ab ab ab ab ab ab LL LL LL LL LL LL LL LL LL ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ = 11 22 33 44 55 66 77 88 99 ab ab ab ab ab ab ab ab ab L L L L L L L L L ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩
=
min{a,b} min{a,b} min{a,b} min{a,b} min{a,b} min{a,b} min{a,b} min{a,b} min{a,b}
11 22 33 44 55 66 77 88 99
Wemakeanoteofthefollowing. Thismin= ∩ (ormax= ∪)aredefinedonsupercolumn vectorsofordinarylabelsonlywhentheyareofsamenatural orderandenjoythesametypeofpartition.
THEOREM
61
L L L L L L L L L .
L L L
2.8: Let K = 1 2 3 4 5 6 7 1 i n n a a a a a a a a a a
L L L L L L L ∈ L = {0 = L0 < L1 < L2 < ... < Lm < Lm+1 = 1}} be the collection of all super column vectors of ordinary labels which enjoy the same form of partition.
(i) {K, ∪} is a semigroup of finite order which is commutative ( semilattice of finite order)
(ii) {K , ∩} is a semigroup of finite order which is commutative ( semilattice of finite order)
(iii) {K , ∪, ∩} is a lattice of finite order.
Nowwestudysupermatricesofordinarylabels.
Consider H=
LL LL LL LL isasupermatrixofordinarylabels. Let M=
aa aa aa aa
12 34 56 78
aa aa aa aa
12 34 56 78
LL LL LL LL againasupermatrixofordinarylabelswithentriesfromL={0 =L0<L1<L2<...<Lm<Lm+1=1}.
Take P=
LL LL LL LL
aa aa aa aa
12 34 56 78
,
Pisagainasupermatrixofordinarylabels.
62
R=
LL LL LL LL isasupermatrixofordinarylabels.
aa aa aa aa
12 34 56 78
V=
LL LL LL LL
aa aa aa aa
12 34 56 78
andW=
LL LL LL LL aresupermatricesordinarylabels.
aa aa aa aa
12 34 56 78
T=
LL LL LL LL
aa aa aa aa
12 34 56 78
andB=
LL LL LL LL aresupermatricesofordinarylabels.
aa aa aa aa
12 34 56 78
C=
LL LL LL LL
aa aa aa aa
12 34 56 78
andD=
LL LL LL LL aresupermatricesofordinarylabels.
aa aa aa aa
12 34 56 78
E=
LL LL LL LL
aa aa aa aa
12 34 56 78
andF=
LL LL LL LL aresupermatricesofordinarylabels.
aa aa aa aa
12 34 56 78
63
G=
LL LL LL LL
aa aa aa aa
12 34 56 78
andL=
LL LL LL LL aresupermatricesofordinarylabels. Now Q=
aa aa aa aa
12 34 56 78
LL LL LL LL
aa aa aa aa
12 34 56 78
isagainasupermatrixofordinarylabels.ThuswehaveH,M, P,R,V,W,T,B,C,D,E,F,G,LandQ;15super matricesofordinarylabelsofnaturalorder4 × 2.
Nowwecanalsoget15supermatrixsemigroupsunder‘∪’ and15supermatrixsemigroupunder ∩ offiniteorder.
Thusbyconvertinganordinarylabelmatricesintosuper matricesmakesonegetseveralmatricesoutofasinglematrix.
THEOREM 2.9: Let A = {all m × n super matrices of ordinary labels of same type of partition}. A is a semigroup under ∩ (or semigroup under ∪) of finite order which is commutative.
Nowhavingseenexamplesandtheoremsaboutsuper matricesofordinarylabels,weinthechapterfourproceedonto definelinearalgebraandvectorspacesofsupermatrices.Itis importanttonotethatingeneralusualproductsarenotdefined amongsupermatrices.Inchapterthreewedefineproductsand transposeofsuperlabelmatrices.Howevereventhecollection ofsupermatricesofsametypearenotclosedingeneralunder product.
64
Inthischapterweproceedontodefinetransposeofasuper matrixofrefinedlabels(ordinarylabels)anddefineproductof supermatricesofrefinedlabels.
beasupercolumnmatrixofrefinedlabelswithentriesfromLR. ClearlythetransposeofAdenotedby
65
L L L L L L L
Let A= 1 2 3 4 5 6 7 a a a a a a a
ThusweseethetransposeofAisthesuperrowvector(matrix) ofrefinedlabels.Ifwereplacetherefinedlabelsbyordinary labelsalsotheresultsholdsgood. Consider P= () 12345678910 aaaaaaaaaa LLLLLLLLLL beasuperrowvector(matrix)ofrefinedlabels. Now i a L ∈ LR;1 ≤ i ≤ 9.NowthetransposeofPdenoted byPtis () 12345678910
t aaaaaaaaaa LLLLLLLLLL=
a a a a a a a a a a
1 2 3 4 5 6 7 8 9 10
weseeP tisasupercolumnmatrix(vector)ofrefinedlabels. Howeverifwereplacetherefinedlabelsbyordinarylabelsstill theresultsholdgood.
66
At = 1 2 3 4 5 6 7 t a a a a a a a L L L L L L L = 1234567 aaaaaaa LLLLLLL;
L L L L L L L L L L ;
67
Let A= 12345 678910 1112131415 1617181920 2122232425 aaaaa aaaaa aaaaa aaaaa aaaaa LLLLL LLLLL LLLLL LLLLL LLLLL beasupermatrixofrefinedlabelswithentriesfromLR;iei a L ∈ LR;1 ≤ i ≤ 25. At = 12345 678910 1112131415 1617181920 2122232425 t aaaaa aaaaa aaaaa aaaaa aaaaa LLLLL LLLLL LLLLL LLLLL LLLLL = 16111621 27121722 38131823 49141924 510152025 aaaaa aaaaa aaaaa aaaaa aaaaa LLLLL LLLLL LLLLL LLLLL LLLLL isagainasupermatrixofrefinedlabelswithentriesfromLR; i a L ∈ LR;1 ≤ i ≤ 25. Let P= 123456 789101112 131415161718 192021222324 252627282930 313233343536 aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL
beasupermatrixofrefinedlabelswithentriesfromLR;i a L ∈ LR;1 ≤ i ≤ 36.
LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL =
Pt =
t aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa
123456 789101112 131415161718 192021222324 252627282930 313233343536
LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL
aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa
1713192531 2814202632 3915212733 41016222834 51117232935 61218243036
isagainasupermatrixofrefinedlabelswithentriesfromLR; i a L ∈ LR;1 ≤ i ≤ 36. Suppose X=
LLL LLL LLL LLL LLL LLL LLL LLL LLL
aaa aaa aaa aaa aaa aaa aaa aaa aaa
123 456 789 101112 131415 161718 192021 222324 252627
68
69
beasupercolumnvector(matrix);withentriesfromLR(i a L ∈ LR;1 ≤ i ≤ 27). ThetransposeofXdenotedby Xt = 123 456 789 101112 131415 161718 192021 222324 252627 t aaa aaa aaa aaa aaa aaa aaa aaa aaa LLL LLL LLL LLL LLL LLL LLL LLL LLL = 147101316192225 258111417202326 369121518212427 aaaaaaaaa aaaaaaaaa aaaaaaaaa LLLLLLLLL LLLLLLLLL LLLLLLLLL isasuperrowvector(matrix)ofrefinedlabelswithentriesfrom LR. Let M= 171319253137 281420263238 391521273339 4101622283440 5111723293541 6121824303642 aaaaaaa aaaaaaa aaaaaaa aaaaaaa aaaaaaa aaaaaaa LLLLLLL LLLLLLL LLLLLLL LLLLLLL LLLLLLL LLLLLLL beasuperrowvector(matrix)ofrefinedlabelswithentries fromLR.
Mt =
LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL
aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa
123456 789101112 131415161718 192021222324 252627282930 313233343536 373839404142
isthetransposeofMwhichisasupercolumnvector(matrix)of refinedlabelswithentriesfromLR.
Nowwefindthetransposeofsupermatrixofrefinedlabels whicharenotsuperrowvectorsorsupercolumnvectors. Let P=
aaaaa aaaaa aaaaa aaaaa aaaaa
12345 678910 1112131415 1617181920 2122232425
LLLLL LLLLL LLLLL LLLLL LLLLL where i a L ∈ LR;1 ≤ i ≤ 25beasupermatrixofrefinedlabels. Pt =
LLLLL LLLLL LLLLL LLLLL LLLLL
aaaaa aaaaa aaaaa aaaaa aaaaa
12345 678910 1112131415 1617181920 2122232425
isthetransposeofPandisagainasupermatrixofrefined labels.
Nowwewillshowhowproductisdefinedincaseofsuper matrixofrefinedlabels.Wejustmakeamentionthatifinthe supermatrixtherefinedlabelsarereplacedbyordinarylabels stilltheconceptoftransposeofsupermatricesremainthesame
70
71
A
=
L L L L L
L L L L L =
12 2
L LL L +
5
L =
+ +
++ =22 12
LL ++ + +222
+++ ++ =22 12
L +++ +222 345
L +++++ =22 12
L ++ +222 345
L +++ =22222 12345
L +++++ .
howevertherewillbedifferenceinthenotionofproductwhich willbediscussedinthefollowing: SupposeA=12345 aaaaa LLLLLbeasuper matrix(rowvector)ofrefinedlabelsthenwefind
t
1 2 3 4 5 a a a a a
, whichisasupercolumnvectorofrefinedlabels. Wefind AAt = 12345 aaaaa LLLLL 1 2 3 4 5 a a a a a
1
a aa a
3 3454
a aaaa a L LLLL
1122 aaaa L.LL.L
334455 aaaaaa L.LL.LL.L
a/(m1)a/(m1)
345 a/(m1)a/(m1)a/(m1) LLL
a/(m1)a/(m1)
a/(m1)a/(m1)a/(m1)
aa/(m1)
aaa/(m1)
aaaaa/(m1)
ClearlyAAt ∈ LRbutitisnotasuperrowvectormatrix. NowweseeifwewanttomultiplyasuperrowvectorAwitha
supercolumnvectorBofrefinedlabelsthenweneedthe following.
(i)IfAisofnaturalorder1 × nthenBmustbeof naturalordern × 1.
(ii)IfAispartitionedbetweenthecolumnsa iand a i+1thenBmustbepartitionedbetweenthe rowsbiandbi+1ofBthismustbecarriedoutfor every1 ≤ i ≤ n.
Wewillillustratethissituationbysomeexamples.
LetA= 123456 aaaaaa LLLLLLbeasuperrow
vectorofnaturalorder1 × 6.ConsiderB=
L L L L L L
b b b b b b
1 2 3 4 5 6
,asuper
columnvectorofnaturalorder6 × 1.ClearlyAispartitioned betweenthecolumns2and3andBispartitionedbetweenthe rows2and3andAispartitionedbetweenthecolumns5and6 andBispartitionedbetweentherows5and6.
Nowwefindtheproduct
AB= 123456 aaaaaa LLLLLL
L L L L L L
b b b b b b
1 2 3 4 5 6
72
L LLLL L + 66 abLL × = 1122 abab L.LL.L + + 334455 ababab L.LL.LL.L ++ 66abL.L = 1122 ab/(m1)ab/(m1)LL++ + 334455 ab/(m1)ab/(m1)ab/(m1) LLL+++ ++ + 66 ab/(m1) L + = 1122 abab/(m1) L ++ + 334455 (ababab)/(m1) L +++ + 66 ab/(m1) L + = 112233445566 (abababababab)/(m1) L ++++++ isinLR. Suppose
L LL L + 3 3454 5 L L L L L L L L L
A= 1 2 3 4 5 6 7 8 9 a a a a a a a a a
and B= 123456789
b aaab b aaaaaaaaa
73
= 1 12 2
b aa b
LLLLLLLLL besupercolumnvectorandsuperrowvectorrespectivelyof refinedlabels.BAisnotdefinedthoughbothAandBareof naturalorder9 × 1and1 × 9respectivelyastheyhavenotthe sametypeofpartitionsforweseeAispartitionedbetweenrow threeandfourbutBispartitionedbetweenoneandtwo columnsandsoon.
74
3 4 5 6 7 8
Letusconsider B= 1234 aaaa LLLLandA= 1 2 3 4 b b b b L L L L besuperrowvectorandsupercolumnvectorrespectivelyof refinedlabels.Wesee BA= 1234 aaaa LLLL 1 2 3 4 b b b b L L L L = () 1122 abab L.LLL++() 3344 abab L.LLL ++ = () 1122 ab/(m1)ab/(m1)LL+++ + | () 3344 ab/(m1)ab/(m1)LL+++ + = 11223344 (abababab)/m1 L ++++ . Letusnowdefineproductofasupercolumnvectorwitha superrowvectorofrefinedlabels. Let A= 1 2
a a a a a a a a L L L L L L L L andB= 123456 cccccc LLLLLL
75
AB= 1 2 3 4 5 6 7 8 a a a a a a a a L L L L L L L L 123456 cccccc LLLLLL = ()() ()() ()() 111 212234526 333 412434546 555 666 12345 777 888 aaa accacccac aaa accacccac aaa aaa ccccc aaa aaa LLL LLLLLLLLL LLL LLLLLLLL.L LLL LLL LLLLL LLL LLL 6c L
beasupercolumnvectorandsuperrowvectorrespectivelyof refinedlabels.WeseetheproductABisdefinedeventhough theyareofverydifferentnaturalorderanddistinctinpartition. Consider
76
= 111213141516 212223242526 313233343536 414243444546 515253 6162 7172 8182 acacacacacac acacacacacac acacacacacac acacacacacac acacaca acac acac acac LLLLLLLLLLLL LLLLLLLLLLLL LLLLLLLLLLLL LLLLLLLLLLLL LLLLLLL LLLL LLLL LLLL 545556 63646566 73747576 83848586 cacac acacacac acacacac acacacac LLLLL LLLLLLLL LLLLLLLL LLLLLLLL 111213141516 212223242526 313233343536 41424344 ac/m1ac/m1ac/m1ac/m1ac/m1ac/m1 ac/m1ac/m1ac/m1ac/m1ac/m1ac/m1 ac/m1ac/m1ac/m1ac/m1ac/m1ac/m1 ac/m1ac/m1ac/m1ac/m1 LLLLLL LLLLLL LLLLLL LLLLL ++++++ ++++++ ++++++ ++++ = 4546 51525354555 6162636465 7172737475 8182838485 ac/m1ac/m1 ac/m1ac/m1ac/m1ac/m1ac/m1a ac/m1ac/m1ac/m1ac/m1ac/m1 ac/m1ac/m1ac/m1ac/m1ac/m1 ac/m1ac/m1ac/m1ac/m1ac/m1 L LLLLLL LLLLL LLLLL LLLLL ++ +++++ +++++ +++++ +++++ 6 66 76 86 c/m1 ac/m1 ac/m1 ac/m1 L L L + + + +
isasupermatrixofrefinedlabels.Wecanmultiplyanysuper columnvectorwithanyothersuperrowvectorandtheproduct isdefinedandtheresultantisasupermatrixofrefinedlabels.
labels.
77
Let C= 1 2 3 4 5 6 7 8 a a a a a a a a L L L L L L L L beasupercolumnvectorofrefinedlabels. Ct = () 12345678 aaaaaaaa LLLLLLLLbethe transposeofC.ClearlyCtisasuperrowvectorofrefined
Wenowfind CCt = 1 2 3 4 5 6 7 8 a a a a a a a a L L L L L L L L () 12345678 aaaaaaaa LLLLLLLL
78 ()()() ()()()()
LLLLLLLLLLLL
()()() 111231456178 2222 12345678 3333 444 515235456 666 aaaaaaaaaaaa aaaa aaaaaaaa aaaa aaaa aaaaaaaaa aaa
LLLL LLLLLLLL LLLL LLLL LLLLLLLLL LLL = () ()()()() 4 578 6 7777 12345678 8888 aaa a aaaa aaaaaaaa aaaa LLL L LLLL LLLLLLLL LLLL 1112131415161718 2122232425262728 3132333435363738 4152434445464748 51 aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa aa LLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLL LLL = 52535455565758 6162636465666768 7172737475767778 8182838485868788 aaaaaaaaaaaaaa aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa LLLLLLLLLLLLL LLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLL
= P.Pt =
LLLLLLLL
/ m1aa/m1aa/m1aa/m1 aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1 aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1
LLLLLLbethetransposeofP.
3738 4152434445464748 5152535455565758 616263646 1 2 3 4 5 6
a/m1aa/m1 aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1 aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1 aa/m1aa/m1aa/m1aa/m1aa a a a a a a
8182838485868788 123456 aaaaaa LLLLLL
isasupermatrixofrefinedlabels,weseethesupermatrixisa symmetricmatrixofrefinedlabels.Thuswegetbymultiplying thecolumnsupervectorwithitstransposethesymmetricmatrix ofrefinedlabels. Wewillillustratebysomemoreexamples.
79
L
2 112131415161718 2122232425262728 313233343536
a/m1aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1 aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1 aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1a LLL LLLLLLLL
LLLLLLLL ++++ ++++++++ ++++++++
LLLLLLLL LLLLLLL ++++++++ ++++++++ ++++++ Let P= 1 2 3 4 5 6 a a a a a a L L L L L L beasupercolumnvectorofrefinedlabels.Pt = 123456 aaaaaa
LLLLLLLL LLLLLLLL LLLLL ++ ++++++++ ++++++++ ++++ 5666768 7172737475767778 L L L L L L
80
= 111 123456 222 333 412434546 555 612634566 aaa aaaaaa aaa aaa aaaaaaaaa aaa aaaaaaaaa LLL LLLLLL LLL LLL LLLLLLLLL LLL LLLLLLLL.L = 111213141516 212223242526 313233343536 414243444546 515253545556 616263 aaaaaaaaaaaa aaaaaaaaaaaa aaaaaaaaaaaa aaaaaaaaaaaa aaaaaaaaaaaa aaaaaaa LLLLLLLLLLLL LLLLLLLLLLLL LLLLLLLLLLLL LLLLLLLLLLLL LLLLLLLLLLLL LLLLLLL 646566 aaaaa LLLLL 2 11213141516 2 2123242526 2 2 3132343536 3 2 41424345 4 a/m1aa/m1aa/m1aa/m1aa/m1aa/m1 aa/m1aa/m1aa/m1aa/m1aa/m1 a/m1 aa/m1aa/m1aa/m1aa/m1aa/m1 a/m1 aa/m1aa/m1aa/m1aa a/m1 LLLLLL LLLLLL LLLLLL LLLLL ++++++ +++++ + +++++ + ++++ = 46 2 5152535456 5 2 61626364656 /m1aa/m1 aa/m1aa/m1aa/m1aa/m1aa/m1 a/m1 aa/m1aa/m1aa/m1aa/m1aa/m1a/m1 L LLLLLL LLLLLL ++ +++++ + ++++++ isasymmetricsupermatrixofrefinedlabels.
LLLLLL LLLLLL LLLLLL
A= 127101112 348131415 569161718
aaaaaa aaaaaa aaaaaa
beasupermatrixofrowvectorsofrefinedlabels.
At =
aaa aaa aaa aaa aaa aaa
135 246 789 101316 111417 121518
LLL LLL LLL LLL LLL LLL bethetransposethesuperrowvectors.Aofrefinedlabels. Tofind
LLLLLL LLLLLL LLLLLL
A.At = 127101112 348131415 569161718
LLL LLL LLL LLL LLL LLL = 12 135 34 246 56
aaaaaa aaaaaa aaaaaa
aaa aaa aaa aaa aaa aaa
135 246 789 101316 111417 121518
LL LLL LL LLL LL + 7 8789 9
aa aaa aa aaa aa
L LLLL L + 101112101316 131415111417 161718121518
a aaaa a
LLLLLL LLLLLL LLLLLL
aaaaaa aaaaaa aaaaaa
81 Let
LLLLLLLLLLLL LLLLLLLLLLLL LLLLLLLLLLLL
aaaaaaaaaaaa aaaaaaaaaaaa aaaaaaaaaaaa
+++ +++ +++ + 777879 878889 979899
aaaaaa aaaaaa aaaaaa
LLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLL LLLLLLLLLL
LLLLLL LLLLLL LLLLLL 101011111212101311141215101611171218 131014111512131314141515131614171518 16101711181216131714
aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaa aaaaaaaaaa
++++++ +++++++ +++ 1815161617171818 aaaaaaaa LLLLLLLL +++ 22 1213241526 22 31423546 34 22 5162536456
+++ =+++ +++ + 2 77879 2 8789 8 2 97989
LLLLLL LLLLLL LLLLLL
a/m1a/m1aa/m1aa/m1aa/m1aa/m1 aa/m1aa/m1aa/m1aa/m1 a/m1a/m1 aa/m1aa/m1aa/m1aa/m1a/m1a/m1
++++++ ++++ ++ ++++++
LLL LLL LLL
a/m1aa/m1aa/m1 aa/m1aa/m1 a/m1 aa/m1aa/m1a/m1
+ LLLLLLLLL LLLLLLLLL L
+++ ++ + +++ a
222 101112101311141215101611171218 222 131014111512131614171518 131415 a/m1a/m1a/m1aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1 aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1 a/m1a/m1a/m1 +++++++++ ++++++ +++
LL LL LL
++++++ ++++++ 161017111812161317141815222 161718 a/m1aa/m1aa/m1aa/m1aa/m1aa/m1a/m1a/m1a/m1 LLLLLLLL +++++++++ ++++++ 222222 127101112132478101311141215 222222 314287131014111512348131415 516297161017111812536498
aaaaaa/m1aaaaaaaaaaaa/m1 aaaaaaaaaaaa/m1aaaaaa/m1 aaaaaaaaaaaa/m1aaaaaaa
++++++++++++ ++++++++++++ +++++++++
= 161317141815 aaaaa/m1 +++
82
= 112213241526 314233443546 516253645566
L L L
152679101611171218 354689131614171518 222222 569161718
++++++ ++++++ ++++++
aaaaaaaaaaaa/m1 aaaaaaaaaaaa/m1 aaaaaa/m1
.
WeseeAAtisaordinarymatrixofrefinedlabels.ClearlyAAt isasymmetricmatrixofrefinedlabels.Thususingaproductof superrowvectorswithitstransposewecangetasymmetric matrixofrefinedlabels. Consider H=
aa aa aa aa aa aa
12 34 56 78 910 1112
LL LL LL LL LL LL asupercolumnvectorofrefinedlabels. Clearly Ht = 1357911 24681012
aaaaaa aaaaaa
LLLLLL LLLLLL isthetransposeofHwhichisasuperrowvectorofrefined labels. Nowwefind HTH=
LL LL LL LL LL LL
aa aa aa aa aa aa
12 34 56 78 910 1112
LLLLLL LLLLLL
× 1357911 24681012
aaaaaa aaaaaa
83
84
LL
++++++++++++ ++++++++++++
= 12 135 34 246 56 aa aaa aa aaa aa LL LLL LL LLL LL + 797811 1112 81091012 aaaaa aa aaaaa LLLLL LL LLLLL + = 113355123456 214365224466 aaaaaaaaaaaa aaaaaaaaaaaa LLLLLLLLLLLL LLLLLLLLLLLL ++++ ++++ + 779978910 87109881010 aaaaaaaa aaaaaaaa LLLLLLLL LLLLLLLL ++ ++ +11111112 12111212 aaaa aaaa LLLL LLLL 222 135123456 222 214365246 a/m1a/m1a/m1aa/m1aa/m1aa/m1 aa/m1aa/m1aa/m1a/m1a/m1a/m1 LLLLLL LLLLLL ++++++ ++++++ ++++ = ++++ + 22 7978910 22 87109810 a/m1a/m1aa/m1aa/m1 aa/m1aa/m1a/m1a/m1 LLLL LLLL ++++ ++++ ++ ++ + 2 111112 2 121112 a/m1aa/m1 aa/m1a/m1 LL LL ++ ++ = 222222 1357911123456789101112 222222 21436587109121124681012 aaaaaa/m1aaaaaaaaaaaa/m1 aaaaaaaaaaaa/m1aaaaaa/m1
LL
isasymmetricmatrixofrefinedlabelswhichisnotasuper matrixofrefinedlabels. Thusinapplicationswecanmultiplytwosupermatricesof refinedlabelsandgetasymmetricmatrixofrefinedlabels,
85
LLLL
L LLLL L LLLL
likewisewecanmultiplysupermatricestogetbacksuper matrices. Forconsider X= 12 34 56 78 910 1112 aa aa aa aa aa aa LL LL LL LL LL LL andY=134 256 bbb bbb LLL LLL twosupermatricesofrefinedlabels.Nowwefind XY= 12 34 56 78 910 1112 aa aa aa aa aa aa LL LL LL LL LL LL 134 256 bbb bbb LLL LLL = ()() 121234 1 3434256 134 5656 256 7878 1 910910 2 11121112 aaaabb b aaaabbb bbb aaaa bbb aaaa b aaaa b aaaa LLLLLL L LLLLLLL LLL
LLL LLLL
34 56 bb bb LL LL
LLLLLLLLLLLL LLLLLLLLLLLL LLLLLLLLLLLL LLLLLLLLLLLL LLLLLLLLLLLL L
+++ +++ +++ +++ +++ 111122113125114126 bababababab LLLLLLLLLLL +++ =
LLL LLL LLL LLL LLL bb bb
=
abab/m1abab/m1abab/m1 abab/m1abab/m1abab/m1 abab/m1abab/m1abab/m1 abab/m1abab/m1abab/m1 abab/m1abab/m1a YtXt = 12 34 56
112213251426 314233453446 516253655466 718273857486 9110293105
bab/m1 abab/m1abab/m1abab/m1 LLL ++ ++++++ isasupermatrixofrefinedlabels. Wefind aaaaaa aaaaaa
++++++ ++++++ ++++++ ++++++ ++++ 94106 111122113125114126 LL LL 1357911 24681012
LLLLLL LLLLLL LLLLLL LLLLLLLLLL LL LLLLLLLLLL LL
LL aaaaaa bbbbbb aaaaaa bbbbabbaaa aa bbbbabbaaa aa
bb LLLLLL LLLLLL ()()()1357911 121212 24681012 35355357911 13 464664681012 24
86
=
112213251426 314233453446 516253655466 718273857486 911029310594106
abababababab abababababab abababababab abababababab abababababab a
LLLLLLLLLLLL LLLLLLLLLLLL LLLLLLLLLLLL LLLLLLLLLLLL LLLLLLLLLLLL
babababababa babababababa babababababa babababababa babababababa
112213241526 315233543556 416243644566 172819210111212 37583951031151
+++ +++ +++ +++ +++ 2 476849610411612 babababababa LLLLLLLLLLLL +++ =
LLL LLL LLL LLL LL
112213241526 315233543556 416243644566 172819210111212 375839510
baba/m1baba/m1baba/m1 baba/m1baba/m1baba/m1 baba/m1baba/m1baba/m1 baba/m1baba/m1baba/m1 baba/m1baba/m1
++++++ ++++++ ++++++ ++++++ ++++ 311512 476849610411612
L LLL ++ ++++++ M=
baba/m1 baba/m1baba/m1baba/m1
Let aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa
LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL
. WeseeXY=YtXtaresupermatrixofrefinedlabels. Nowweproceedontodefineanothertypeofproductof supermatrixofrefinedlabels. 1815222936 2916233037 31017243138 41118253239 51219263340 61320273441 71421283542
beasupermatrixofrefinedlabels.
87
=
MT =
LLLLLLL LLLLLLL LLLLLLL LLLLLLL LLLLLLL LLLLLLL
aaaaaaa aaaaaaa aaaaaaa aaaaaaa aaaaaaa aaaaaaa
1234567 891011121314 15161718192021 22232425262728 29303132333435 36373839404142
M.MT =
LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL
aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa
1815222936 2916233037 31017243138 41118253239 51219263340 61320273441 71421283542
LLLLLLL LLLLLLL LLLLLLL LLLLLLL LLLLLLL LLLLLLL
aaaaaaa aaaaaaa aaaaaaa aaaaaaa aaaaaaa aaaaaaa
× 1234567 891011121314 15161718192021 22232425262728 29303132333435 36373839404142
88 Consider
89
1
L L L LLLLLLLL L L L +
LL LL LL LLLLLLL LL LLLLLLL LL LL LL
233037 24313822232425262728 25323929303132333435 26334036373839404142 273441 283542
aaa aaa LLL LLL LLLLLLLLLL LLLLLLLLLL LLLLLLLLLL LLL LLL
=
2 3 41234567 5 6 7 a a a aaaaaaaa a a a
815 916 1017 891011121314 1118 15161718192021 1219 1320 1421 aa aa aa aaaaaaa aa aaaaaaa aa aa aa
+ 222936
aaa aaa aaaaaaaaaa aaaaaaaaaa aaaaaaaaaa
=
LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLL LLL LLL L
aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa aaa aaa aaa a
222324252627 222936222936 293031323334 233037233037 363738394041 243138 253239 263340 2
aaa aaaaaa aaa aaaaaa aaa aaaaaa aaaaa a aaa
243138 222324252627 253239 293031323334 263340 363738394041 73441273441 2 282930
LLL LLLLLL LLL LLLLLL LLL LLLLLL LLLLL L LLL 22324252627 293028293031323334 363738394041
aaaaa aaaaaaaaa aaaaaa
L LLL L LLL L LLL L LLL L LLL L LLL
LLLLL LLLLLLLLL LLLLLL 28 222936 35 233037 42 243138 28 253239 35 263340 42 273441 28 28293035 42
a aaa a aaa a aaa a aaa a aaa a aaa a aaaa a
90
L LLLL L
91 + ()() ()() ()() 111
LLL LLLLLLL LLL LLL LLL LLLLLLL LLL LLL
LLLLLLLLLLLLL
LL
LLLL LLLLLLLLL
LL LLLL LL
LLLLL
31323334353637 41424344454647 5152535455
aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaa LLLLLLLLLLLLLL LLLLLLLLLLLLLL LLLLLLLLLLLLLL LLLLLLLLLLLLLL LLLLLLLLLLL = 5657 61626364656667 71727374757677 aaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa LLL LLLLLLLLLLLLLL LLLLLLLLLLLLLL
1234567 222 333 444 1234567 555 666 7127345677 aaa aaaaaaa aaa aaa aaa aaaaaaa aaa aaa aaaaaaaaaa
LLLLLLLLL.L 815898151011121381514 91615169161718192091621 1017 111889 12191 1320 aaaaaaaaaaaaa aaaaaaaaaaaaa aa aaaa aaa aa
LLLLLLLLLLLLL
LLLL LLL LL + 10171017 111810111213111814 516121917181920121921 13201320 89 142114 1516 aaaa aaaaaaaaa aaaaaaaaaa aaaa aa aaa aa
LLLLLLLLLL LLLL
1011121314 211421 1718192021 aaaaa aaa aaaaa
LL LLLLL 11121314151617 21222324252627
aaaaaaaaaaaaaa aaaaaaaaaaaaaa
92
LLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLL LLLLLLLLLL ++++++ ++++++ ++++ 3837242431313838 252232293936252332303937252432313938 262233294036262333304037262433314038 2722 aaaaaaaa aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaa aa LLLLLLLL LLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLL LLL ++ ++++++ ++++++ + 34294136272334304137272434314138 282235294236282335304237282435314238 aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaa LLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLL +++++ ++++++
88151589151681015178111518 98161599161691016179111618 108171510917161010171710111718 118 aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa aa LLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLL LLL ++++ ++++ ++++ ++ 181511918161110181711111818 128191512919161210191712111918 138201513920161310201713112018 1482 aaaaaaaaaaaaaa aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa aaa LLLLLLLLLLLLL LLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLL LLL +++ ++++ ++++ + 11514921161410211714112118 aaaaaaaaaaaaa LLLLLLLLLLLLL +++ 912161981315208141521 912161991316209141621 101217191013172010141721 111218191113182011141821 aaaaaaaaaaaa aaaaaaaaaaaa aaaaaaaaaaaa aaaaaaaaaaaa a LLLLLLLLLLLL LLLLLLLLLLLL LLLLLLLLLLLL LLLLLLLLLLLL L +++ +++ +++ +++ 121219191213192012141921 131220191313202013142021 141221191413212014142121 aaaaaaaaaaa aaaaaaaaaaaa aaaaaaaaaaaa LLLLLLLLLLL LLLLLLLLLLLL LLLLLLLLLLLL +++ +++ +++ + 222229293636222329303637222429313638 232230293736232330303737232530313738 24223129383624233130 aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaa aaaaaaaaaa
93
++++++
222529323639222629333640222729343641 232530323739232630333740232730343741 24253132383924263133 aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaa aaaaaaaaaa LLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLL LLLLLLLLLL ++++++ ++++++ ++++ 3840242731343841 252532323939252632333940252732343941 262533324039262633334040262733344041 2725 aaaaaaaa aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaa aa LLLLLLLL LLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLL LLL ++ ++++++ ++++++ + 34324139272634334140272734344141 282535324239282635334240282735344241 aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaa LLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLL +++++ ++++++ 222829353642 232830353742 242831353842 252832353942 262833354042 272834354142 282835354242 aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL ++ ++ ++ ++ ++ ++ ++ 2 1121314151617 2 212324252627 2 2 313234353637 3 414 a/m1aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1 aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1 a/m1 aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1 a/m1 aa/m1aa LLLLLLL LLLLLLL LLLLLLL LL +++++++ ++++++ +
+ =+ 2 243454647 4 2 515253545657 5 2 616263646567 6 717273 /m1aa/m1aa/m1aa/m1aa/m1 a/m1 aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1 a/m1 aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1 a/m1 aa/m1aa/m1aa/m1 LLLLL LLLLLLL LLLLLLL LLL +++++ + ++++++ + ++++++ + +++ 2 7475767 aa/m1aa/m1aa/m1a/m1 LLLL++++
LLLL LLLL LLLL L
++++++++ ++++++ ++ +++++++ + ++ 22 18151191816111018171118 128191512919161210191712111918 138201513920161310201713112018
LLL LLLL LLLL L
22 81589151681015178111518 22 98161591016179111618 916 22 1081715109171610111718 1017 118
aa/m1aaaa/m1aaaa/m1aaaa/m1 aaaa/m1aaaa/m1aaaa/m1 aa/m1 aaaa/m1aaaa/m1aaaaa/m1 a/m1 aa
aa/m1aaaa/m1aaaa/m1aa/m1 aaaa/m1aaaa/m1aaaa/m1aaaa/m1 aaaa/m1aaaa/m1aaaa/m1aaaa/m1
LLL LLL LLL LLL
aaaa/m1aaaa/m1aaaa/m1 aaaa/m1aaaa/m1aaaa/m1 aaaa/m1aaaa/m1aaaa/m1 aaaa/m1aaaa/m1aa
+++++++ ++++++++ ++++++++ 148211514921161410211714112118 aaaa/m1aaaa/m1aaaa/m1aaaa/m1 LLL ++++++++ 812151981315208141521 912161991316209141621 101217191013172010141721 11121819111318201114
LLL LLL LLL
aa/m1 / aam1aaaa/m1aaaa/m1 aaaa/m1/m1aaaa/m1 aa aaaa/m1aaaa/m1/m1 aa
++++++ ++++++ ++++++ +++++ 1821 22 12191213192012141921 22 1312201913142021 1320 22 14122119141321201421
LLL LLL LLL
+ ++++++ +++++ + +++++ +
+ 222 222936222329303637222429313638 222 232230293736232530313738 233037 22 2422312938362423313038372431
LLL LLL LL
aaa/m1aaaaaa/m1aaaaaa/m1 aaaaaa/m1aaaaaa/m1 aaa/m1 aaaaaa/m1aaaaaa/m1aaa
+++++++++ ++++++ +++ ++++++++ 2 38 252232293936252332303937252432313938 262233294036262333304037262433314038 2722342941362723343041
L LLL ++++ +++++++++
/m1 aaaaaa/m1aaaaaa/m1aaaaaa/m1 aaaaaa/m1aaaaaa/m1aaaaaa/m1 aaaaaa/m1aaaaa
a/m1aaaaaa/m1 aaaaaa/m1aaaaaa/m1aaaaaa/m1
+ +++++++++ +++++++++ +++++ 37272434314138 282235294236282335304237282435314238
94
L LL
L
LLL +++++ +++++++++ 222829353642 232830353742 242831353842 252832353942 262833354042 272834354142 222 283542
LLL LLL LLL +++++ ++++++++++++ +++++++++++
aaaaaa/m1 aaaaaa/m1 aaaaaa/m1 aaaaaa/m1 aaaaaa/m1 aaaaaa/m1 aaa/m1
+++ +++++++++ ++++++ +++ ++++ 222 334140273441 282535324239282635334240282735344241 LL LL
091716242331303837 411181815252232293936421191816252332303937 5112819152622332940365212919162623333040 aaaaaaaaa/m1 aaaaaaaaaaaa/m1aaaaaaaaaaaa/m1 aaaaaaaaaaaa/m1aaaaaaaaaaa LL
aaa/m1aaa/m1 aaaaaa/m1aaaaaa/m1aaaaaa/m1 ++++++++++++ ++++++++++++
37 611382015272234294136621392016272334304137 711482115282235294236721492116282335304237 a/m1 aaaaaaaaaaaa/m1aaaaaaaaaaaa/m1 aaaaaaaaaaaa/m1aaaaaaaaaaaa/m1
+
95
222529323639222629333640222729343641 232530323739232630333740232730343741 242531323839242631333840
aaaaaa/m1aaaaaa/m1aaaaaa/m1 aaaaaa/m1aaaaaa/m1aaaaaa/m1 aaaaaa/m1aaaaaa/m1
LLL LLL LL L L L L L L L
+++++++++ +++++++++ ++++++ 242731343841 222 253241252632333940252732343941 222 262533324039262733344041 263340 272534324139272634 LL LL LL
aaaaaa/m1 aaa/m1aaaaaa/m1aaaaaa/m1 aaaaaa/m1aaaaaa/m1 aaa/m1 aaaaaa/m1aaa ++++++++++++ ++++++++++++ +++++++ =
+++ +++ +++ +++ +++ +++ +++ 222222 181522293612891516222329303637 222222 219816152322302937362916233037 311081715242231293836321 aaaaaa/m1aaaaaaaaaaaa/m1 aaaaaaaaaaaa/m1aaaaaa/m1 aaaaaaaaaaaa/m1aaa
LL LL L
L LL LL
aaaaaaaaaaaa/m1aaaaaaaaaaaa/m1 aaaaaaaaaaaa/m1aaaaaaaaaaaa/m1 aaaaaa
++++++++++++ ++++++++++++ +++++ 2 3410111718242531323839 222222 431110181725243231393841118253239 53121019172624333140385412111918262533
LL LL
/m1aaaaaaaaaaaa/m1 aaaaaaaaaaaa/m1aaaaaa/m1 aaaaaaaaaaaa/m1aaaaaaaaa
+++++++ ++++++++++++ ++++++++++ 324039 63131020172724343141386413112018272534324139 73141021172824353142387414112118282535324239
LL LL L
L LL LL
aaa/m1 aaaaaaaaaaaa/m1aaaaaaaaaaaa/m1 aaaaaaaaaaaa/m1aaaaaaaaaaaa/m1
aaaaaaaaaaaa/m1aaaaaaaaaaaa/m1 aaaaaaaaaaaa/m1aaaaaaaaaaaa/m1 aaaaaaaaa
++ ++++++++++++ ++++++++++++ 158121519222629333640168131520222729343641 259121619232630333740269131620232730343741 35101217192426
++++++++++++ ++++++++++++ ++++ 313338403610131720242731343841 45111218192526323339406611131820252732343941 222222 512192633405612131920
LL LL ++++ ++++++++++++ ++++++++++++ L L L L L
aaa/m1aaaaaaaaaaaa/m1 aaaaaaaaaaaa/m1aaaaaaaaaaaa/m1 aaaaaa/m1aaaaaa
aaaaaa/m1 aaaaaaaaaaaa/m1aaaaaa/m1 aaaaaaaaaaaa/m1aaaaaaaaaaaa/m1 aaaaaaaaaaaa/m1 aaaaaaaaaaaa/m1 aaaaaaaaaaaa/m1 aaaaaaaaaaaa/m1 aaaaaaaa
++++++++ ++++++++++++ ++++++++ 262733344041 222222 651312201927263433414061320273441 75141221192826353342407614132120282735344241 178141521222829353642 279141621232830353742 3710141721242831353842 4711141821252832353942 57121419212628
L L
++++++ ++++++ ++++++ ++++++ +++ 33354042 6713142021272834354142 222222 71421283542
. Weseetheproductisagainsupermatrixofrefinedlabels butisclearlyasymmetricsupermatrixofrefinedlabelsof naturalorder7 × 7.Thusbythistechniqueofproductwecan getsymmetricsupermatrixofrefinedlabelswhichmayhave applicationsinrealworldproblems.
aaaa/m1 aaaaaaaaaaaa/m1 aaaaaa/m1
+++ ++++++ ++++++
96
138101517222429313638148111518222529323639 239101617232430313738249111618232530323739 22222 31017243138
Weseeattimestheproductoftwosupermatricesofrefined labelsmayturnouttobejustmatricesorsymmetricsuper matricesofrefinedlabelsdependingonthematricesproducted andtheproductsdefined[48].
NowweproceedontodefinethenotionofM–matrixof labelsZ–matriceslabelsandthecorrespondingsubdirectsums.
LetA= () {} ii aaR mn LLL × ∈ beam × nrefinedlabelmatrix
wesayA>0ifeachi a L=i a m1 + >0.(A ≥ 0ifeach i a L= 1 ai m + ≥ 0).ThusifA ≥ BweseeA–B ≥ 0.Squarematrices withrefinedlabelswhichhavenonpositiveoff–diagonal entriesarecalledZ-matricesofrefinedlabels.
Wesee–La=L-a= a m1 + . A= 123 456 789
aaa aaa aaa
LLL LLL LLL isaZ-matrixofrefinedlabels wherea i>0.
WecallaZ-matrixofrefinedlabelsAtobeanonsingular M-matrixofrefinedlabelsifA-1 ≥ 0.
WehavefollowingpropertiestobetrueforusualMmatriceswhichisanalogouslytrueincaseofM-matricesof refinedlabels.
(i)ThediagonalofanonsingularM-matrixofrefined labelsispositive.
(ii)IfBisaZ-matrixofrefinedlabelsandMisanon singularM-matrixandM ≤ BthenBisalsoanon singularM-matrix.
97
InparticularanymatrixobtainedfromanonsingularMmatrixofrefinedlabelsbysettingoff-dia gonalentriesiszerois alsoanonsingularM-matrix. Thus A= 123 456 789
aaa aaa aaa
LLL LLL LLL ;1 a L=1 a m1 + >0, 5 a L=5 a m1 + >0and 9 a L=9 a m1 + >0 2 a L=2 a m1 + <0,3 a L=3 a m1 + <0,4 a L=4 a m1 + <0, 6 a L=6 a m1 + <0, 7 a L=7 a m1 + <0and 8 a L=8 a m1 + <0.
AisanonsingularM-matrixandthediagonalelementsare positive. AmatrixMisnonsingularM-matrixifandonlyifeach principalsubmatrixofMisanonsingularM-matrix. Forinstancetake A= 123 456 789
aaa aaa aaa
LLL LLL LLL tobeanonsingularM-matrixthen1 a L ≠ 0 ∈ LRand 56 89
aa aa
LL LLarenonsingulari a L>0 thatisi a m1 + >0.1 ≤ i ≤ 9and i a L ∈ LR.
98
FinallyaZ-matrixAisnonsingularM-matrixifandonlyif thereexistsapositivevectorLx>0suchthatALx>0. ThusallresultsenjoyedbysimpleM-matricescanbe derivedanalogouslyforM-matricesofrefinedlabels.
Nowwecanrealizethesestructuresassupermatricesofrefined labels.
Suppose A= 123 456 789
aaa aaa aaa
LLL LLL LLL and B= 123 456 789
LLL LLL LLL 1b L= 5 a Lbe 6 a L= 2 b L 9 a L= 5 b Land 4 b L= 8 a L.
bbb bbb bbb
TwosupermatricesofrefinedlabelsthenA ⊕2B=Cisnota M-matrixofrefinedlabels. C=
LLL0 LLLL LLLL 0LLL
123 4123 5456 789
aaa a2b2bb a2b2bb bbb
WecanfindC-1.WeseeCissupermatrixofrefinedlabels andthesumisnotausualsumofsupermatrices. Thismethodofadditionofoverlappingsupermatricesof refinedlabelsmayfinditselfusefulinapplications.Allresults studiedcanbeeasilydefinedformatricesofrefinedlabelswith simpleappropriateoperations.
99
WecanalsodefinetheP-matrixofrefinedlabelsasincase ofP-matrix.
Furtheritisleftasanexerciseforthereadertoprovethatin generalk-subdirectsumoftwoP-matricesofrefinedlabelsis notaP-matrixofrefinedlabelsingeneral. If A= 123 45 67
aaa aa aa
L0L 0LL LLL
bb bb bbb
LLL LL0 L0L andB= 12 34 767
betwoP-matricesofrefinedlabelsthenwehavethe2-subdirect sumas C=A ⊕2B=
aaa aabb aabb bbb
LLL0 LL0L L0LL 0LLL
123 4512 6734 567
+ +
isnotaP-matrixofrefinedlabelsasdet(C)<0.
ThustheconceptofM-matrices,Z-matricesandP-matrices canalsoberealizedasthesupermatricesofrefinedlabels.Also thek-subdirectsumisalsoagainasupermatrixbutadditionis different.Forinusualsupermatricesadditioncanbecarried outforwhichtheresultantafteradditionenjoythesame partition.Hereweseethek-subdirectsumisdefinedonlyin thosecasesofdifferentpartitionandhoweverthek-subdirect sumofsupermatricesdonotenjoyeventhesamenaturalorder asthatofthetheircomponents.Thustheydonotformagroup orasemigroupunderk-subdirectaddition.
100
Inthischapterweforthefirsttimeintroducethenotionof supervectorspaceofrefinedlabelsusingthesupermatricesof refinedlabels.Byusingthenotionofsupervectorspaceof refinedlabelsweobtainmanyvectorspacesasagainstusual matricesofrefinedlabels.Weproceedontoillustratethe definitionsandpropertiesbyexamplessothatthereadergetsa completegraspofthesubject.
LL;1in } be the collection of all super row vectors of same type of refined labels. V is an additive abelian group. V is a super row vector space of refined labels over R or LR (as LR ≅ R).
Wewillillustratethissituationbysomeexamples.
101
(){ ++ 12345rr1r2n aaaaaaaaa LLLLL...LLL...L
i
DEFINITION 4.1 : Let V =
∈≤≤
aR
Example 4.1:Let
V= () {} 123456i aaaaaaaR LLLLLLLL;1i6 ∈≤≤ beasuperrowvectorspaceofrefinedlabelsoverthefieldLR (orR).
Example 4.2:Let V= () {} 123456i aaaaaaaR LLLLLLLL;1i6 ∈≤≤ bea superrowvectorspaceofrefinedlabelsoverthefieldLR(orR).
Example 4.3:Let W= () {} 123456i aaaaaaaR LLLLLLLL;1i6 ∈≤≤ bea superrowvectorspaceofrefinedlabelsoverthefieldLR(orR).
Example 4.4:Let V= () {} 123456i aaaaaaaR LLLLLLLL;1i6 ∈≤≤ bea superrowvectorspaceofrefinedlabelsoverthefieldLR(orR).
Weseeforagivenrowvector () 123456 aaaaaa LLLLLLofrefinedlabelswecanget severalsuperrowvectorspaceofrefinedlabelsoverthefield LRorR.Thusthisisoneofthemainadvantageofdefining superrowvectorsofrefinedlabelsoverLRorR.
Example 4.5:Let X= (){ 12345678910 aaaaaaaaaa LLLLLLLLLL i aR LL;1i10 ∈≤≤ }beacollectionofallsuperrowvectorsof refinedlabels.XisagroupunderadditionandXisasuperrow vectorspaceofrefinedlabelsoverLR(orR).
Nowhavingseenexamplesofsuperrowvectorspacesof refinedlabelswenowproceedontodefinesubstructuresin them.
102
103
: Let V = (){ 1234567n1n aaaaaaaaa LLLLLLL...LL ;1 ∈≤≤ i aR
} be a
P= () {} 1234567i aaaaaaaaR LLLLLLLLL;1i7 ∈≤≤ beasupervectorspaceofVoverthefieldL
Consider S= () {} 1234i aaaaaR L0L0L0LLL;1i7∈≤≤⊆ P isasuperrowvectorsubspaceofrefinedlabelsofPoverthe
Take T= () {} 123i aaaaR 0L0L0L0LL;1i3
P,
4.7: Let X= () {} 1234i aaaaaR LLLLLL;1i4 ∈≤≤ beasuperrowvectorspaceofrefinedlabelsoverL
Consider X1= () {} 1i aaR L000LL∈⊆ X, X2= () {} 2i aaR 0L00LL∈⊆ X, X3= () {} 3i aaR 00L0LL∈⊆ X and X4= () {} 4i aaR 000LLL∈⊆ X befoursuperrowvectorsubspacesofrefinedlabelsofXover LR.
DEFINITION 4.2
LLin
super row vector space of refined labels over the field LR. Suppose W ⊆ V; W a proper subset of V; if W is a super row vector space of refined labels over LR. then we define W to be super vector subspace of V over the field LR Example 4.6:Let
R(orR).
fieldLR(orR).
∈≤≤⊆
TisasuperrowvectorsubspaceofrefinedlabelsofP. Example
R(orR).
WeseeX= 4 1 i i X =
andXi ∩ Xj=(0|000)ifi ≠ j;1 ≤ i,j ≤ 4.
HoweverXhasmorethanfoursuperrowvectorsubspaces ofrefinedlabels.
FortakeY1= () {} 12i aaaR LL00LL;1i2∈≤≤⊆ Xis asuperrowvectorsubspaceofrefinedlabels.
ConsiderY2= () {} 12i aaaR L00LLL;1i2∈≤≤⊆ Xis asuperrowvectorsubspaceofXofrefinedlabels.
TakeY3= () {} 12i aaaR 0LL0LL;1i2∈≤≤⊆ Xis
againasuperrowvectorsubspaceofXofrefinedlabelsover LR.
DEFINITION 4.3: Let V be a super row vector space of refined labels over the field LR (or R). Suppose W1, W2, …, Wm are super row vector subspaces of refined labels of V over the field LR (or R). If V = 1
m i i W =
and Wi ∩ Wj = (0) if i≠j, 1 ≤ i, j ≤ m then we say V is a direct union or sum of super row vector subspaces of V.
If W1, …, Wm are super row vector subspaces of refined labels of V over LR (or R) and V = 1
m i i W =
but Wi ∩ Wj ≠ (0) if i≠j, 1 ≤ i, j ≤ m then we say V is a pseudo direct sum of super vector subspaces of refined labels of V over LR (or R).
Example 4.8:Let V= () {} 1234567i aaaaaaaaR LLLLLLLLL;1i7 ∈≤≤ beasuperrowvectorspaceofrefinedlabelsoverLR(orR). Consider W1= () {} 12i aaaR LL00000LL;1i2∈≤≤⊆ V,
104
105 W2= () {} 1234i aaaaaR L0LL00LLL;1i4∈≤≤⊆ V, W3= () {} 134i aaaaR L00LL00LL;1i3∈≤≤⊆ V, W4= () {} 12435i aaaaaaR L0L0LLLLL;1i5∈≤≤⊆ Vand W5= () {} 125i aaaaR 0LL000LLL;1i3∈≤≤⊆ V besuperrowvectorsubspacesofrefinedlabelsofVoverLR. ClearlyV= 5 1 i i W = butWi ∩ Wj ≠ (00|0|000|0)ifi≠ j,1 ≤ i, j ≤ 5. ThusVisonlyapseudodirectsumofW
Example 4.9:LetV= () {} 12356784i aaaaaaaaaR LLLLLLLLLL;1i8 ∈≤≤ beasupervector. Consider W1= () {} 1i aaR L0000000LL∈⊆ V, W2= () {} 1i aaR 0L000000LL∈⊆ V, W3= () {} 1i aaR 00L00000LL∈⊆ V, W4= () {} 1i aaR 000L0000LL∈⊆ V, W5= () {} 1i aaR 0000L000LL∈⊆ V, W6= () {} 1 0000000 ∈ i aaR LLL ⊆ V, W7= () {} 1i aaR 000000L0LL∈⊆ Vand
i,i=1,2,…,5and isnotadirectsum.Nowweseethesupervectorspaceof refinedlabelsasincaseofothervectorspacescanbewritten bothasadirectsumaswellaspseudodirectsumofsuper vectorsubspacesofrefinedlabelsoverLR(orR).
W8= () {} 1i aaR 0000000LLL∈⊆ V; aresuperrowvectorsubspacesofVoverthefieldLR.Clearly V= 8 i i1 W =
butWi ∩ Wj=(00|00|0000)ifi≠ j,1 ≤ i,j ≤ 8. VisadirectsumofsuperrowvectorsubspacesW1,W2,…, W8ofVoverLR. Consider W1= () {} 123i aaaaR LL00000LLL;1i3∈≤≤⊆ V, W2= () {} 123i aaaaR L0LL0000LL;1i3∈≤≤⊆ V, W3= () {} 123i aaaaR 0L00LL00LL;1i3∈≤≤⊆ V and W4= () {} 1234i aaaaaR L0L0L0L0LL;1i3∈≤≤⊆ V besuperrowvectorsubspacesofVovertherefinedfieldLR(or R).ClearlyV= 4 i i1 W = isapseudodirectsumofsuperrow vectorsubspacesofVoverthefieldLRasWi ∩ Wj ≠ (00|00| 0000)ifi ≠ j,1 ≤ i,j ≤ 4. Nowhavingseenexamplesofdirectsumandpseudodirect sumofsuperrowvectorsubspacesofVoverLRwenow proceedontodefinelinearoperators,basisandlinear transformationofsuperrowvectorspacesofrefinedlabelsover LR.
ItispertinenttomentionherethatLR ≅ Rsowhetherwe workonRorLRitisthesame.
106
t a1a2a LxLx...Lx0 . A set which is not linearly dependent is called linearly independent. We say for the super row vector space of refined labels V = () {} ∈≤≤ 12345n1ni aaaaaaaaR LLLLL...LLLL;1in a subset B of V to be a basis of V if (1) B is a linearly independent subset of V. (2) B generates or spans V over LR. We will give examples of them. It is pertinent to mention in LR; Lm+1 is the unit element of the field LR. Example 4.10: LetV= () {} 1234567i aaaaaaaaR LLLLLLLLL;1i7 ∈≤≤ bea superrowvectorspaceofrefinedlabelsoverthefieldLR.
Considertheset B={(Lm+1,0|0|000|0),(0,Lm+1|0|000|0),(00| Lm+1|000|0),(00|0|Lm+100|0),(00|0|0Lm+10|0),(0 0|0|00Lm+1|0),(00|0|000|Lm+1)} ⊆ Visasuperbasisof overLR.ClearlyVisfinitedimensionaloverLR. ConsiderP= () 127 aaa LL0000L,
107
()
()
DEFINITION 4.4 : Let V = () {} ∈≤≤ 12n1ni aaaaaR LL...LLLL;1in be a super row vector space of refined labels over the field LR. We say a subset of elements {x1, x2, …, xt} in V are linearly dependent if there exists scalar labels 1t aa L...L in LR not all of them equal to zero such that +++= 12t ⊆ V;Pisonlyasuperlinearly
2 a 00L0000, () 3 a 000L000,
45 aa 0000LL0
independentsubsetofVbutisnotabasisofV.
linearlyindependentsubsetandnotabasisandlinear independentsubsetofVwhichisasuperbasisofV,wenow proceedontodefinethelinearlytransformationofsuperrow vectorspacesofrefinedlabelsoverL
108
Example 4.11: LetV= () {} 123i aaaaR LLLLL;1i3; ∈≤≤ beasuperrowvectorspaceofrefinedlabelsoverLR.Consider B={(Lm+1|00),(0,|Lm+10),(0|0Lm+1)} ⊆ Visasuperbasis ofVoverLR. ClearlydimensionofVisthreeoverLR.ConsiderM = ()() {} 112 aaa L00,0LL ⊆ V;Misonlyalinearly independentsetofVoverLRandisnotabasisofVoverLR. TakeK= ()()(){ 121212 aaaaaa L0L,LL0,0LL, ()()} 12 aa L00,0L0 ⊆ V,Kisalinearlydependent subsetofV.ConsiderB= ()()() {} 1122 aaaa 0L0,0LL,00L ⊆ V.Bisalsoa linearlydependentsubsetofVoverLR. Nowhavingseenexamplesoflinearlydependentsubset,
R LetVandWbetwosuperrowvectorspaceofrefined labelsoverLR.LetTbeamapfromVtoWwesayT:V → W isalineartransformationifT(cv+u)=cT(v)+T(u)forallu, v ∈ Vandc ∈ LR. Example 4.12: LetV= () {} 123456i aaaaaaaR LLLLLLLL;1i6 ∈≤≤ beasuper rowvectorspaceofrefinedlabelsoverLR.W= () {} 12345678i aaaaaaaaaR LLLLLLLLLL;1i8 ∈≤≤ beasuperrowvectorspaceofrefinedlabelsoverLR.Definea mapT:V → WbyT () 123456 aaaaaa LLLLLL= () 123456 aaaaaa L0L0LLLL0;Tiseasily verifiedtobealineartransformation.
109 LetP:V → Wbeamapsuchthat () 123456 aaaaaa LLLLLL= () 123456 aaaaaa LLLLLL00;ItiseasilyverifiedP isalineartransformationofsuperrowvectorspaceVintoW. Example 4.13: LetV= (){ 12345678910 aaaaaaaaaa LLLLLLLLLL } i aR LL;1i10 ∈≤≤ beasuperrowvectorspaceofrefined labelsoverthefieldLR.LetTbeamapfromVtoVdefinedby T () 12345678910 aaaaaaaaaa LLLLLLLLLL= () 13579 aaaaa L0L0L0L0L0;Tisalinear operatoronV.ItisclearTisnotonetoone.KerTisanon zerosubspaceofV.ConsideramapP:V → Vdefinedby P () () 12345678910 aaaaaaaaaa LLLLLLLLLL= () 32154610987 aaaaaaaaaa LLLLLLLLLL.Pis alinearoperatorwithtrivialkernel. NowsupposeV= () {} 12345678i aaaaaaaaaR LLLLLLLLLL;1i8 ∈≤≤ beasuperrowvectorspaceofrefinedlabelsoverthefieldLR. ConsiderW= () {} 1238i aaaaaR LLL0000LLL;i1,2,3,8∈=⊆V; WisasuperrowvectorsubspaceofVofrefinedlabelsoverthe fieldLR. LetT:V → VifT(W) ⊆ W,WisinvariantunderT,we canillustratethissituationbysomeexamples.
isalinear transformationandMisinvariantunder η as η (M) ⊆ M. Asincaseofusualvectorspacesincaseofsuperrowvector spaceofrefinedlabelsoverthefieldLRwecanhavethe followingtheorem.
THEOREM 4.1: Let V and W be two finite dimensional super vector spaces of refined labels over the field LR such that dim V = dim W. If T is a linear transformation from V into W, the following are equivalent.
(i) T is invertible (ii) T is non singular (iii) T is onto, that is range of T is W.
Theproofisleftasanexercisetothereader. Furtherwecanasincaseofusualvectorspacedefinein caseofsuperrowvectorspaceofrefinedlabelsalsothe followingresults.
THEOREM 4.2: If {α1, α2, …, αn} is a basis for V then {T α1, T α2, …, T αn} is a basis for W. (dim V = dim W = n is assumed), V a finite dimensional super vector space over LR and Ta is a linear transformation from V to W.
Thisproofisalsodirectandhenceleftasanexercisetothe reader.
110 T () ()
aaaaaaaa
12345678
LLLLLLLL= () 1238 aaaa LLL0000L;issuchthatT(W)⊆ W. SupposeM= () {} 1234i aaaaaR L0L0L0L0LL;1i4∈≤≤⊆ V; thenMisasuperrowvectorsubspaceofVofrefinedlabels overLR.Define η :V → Vby η() () 12345678 aaaaaaaa LLLLLLLL= () 1234 aaaa L0L0L0L0; η
THEOREM 4.3: Let V and W be two super row vector spaces of same dimension over the real field LR. There is some basis {α1, α2, …, αn} for V such that {T α1, Tα2, …, T αn} is a basis for W. Theproofofthisresultsisdirect.
THEOREM 4.4: Let T be a linear transformation from V into W, V and W super row vector spaces of refined labels over the field LR then T is non singular if and only if T carries each linearly independent subset of V onto a linearly independent subset of W.
Wemakeuseofthefollowingtheoremtoprovethetheorem 4.4.
THEOREM 4.5: Let V and W be any two super row vector spaces of refined labels over the field LR and let T be a linear transformation from V into W. If T is invertible then the inverse function T -1 is a linear transformation from W onto V.
Alsoasincaseofusualvectorspaceweseeincaseofsuper rowvectorspaceVofrefinedlabelsoverafieldF=LRifT 1,T 2 andParelinearoperatorsonVandforc ∈ LRwehave
(i)IP=PI=P (ii)P(T 1+T 2)=PT 1+PT 2 (iii)(T 1+T 2)P=T 1P+T 2P (iv)c(PT 1)=(cP)T 1=P(cT 1).
Alltheresultscanbeprovedwithoutanydifficultyand henceisleftasanexercisetothereader.
AlsoifV,WandZaresuperrowvectorspacesofrefined labelsoverthefieldLRandT:V → Wbealinear transformationofVintoWandPalineartransformationofW intoZ,thenthecomposedfunctionPTdefinedby(PT)(α)= P(T(α))isalineartransformationfromVintoZ.Thisproofis alsodirectandsimpleandhenceisleftasanexercisetothe reader.
Nowhavingstudiedpropertiesaboutsuperrowvector spacesofrefinedlabelswenowproceedontodefinesuper columnvectorspacesofrefinedlabelsoverLR.
111
L L L LL1in L L
1 2 3 i n1 n
a a a aR a a
DEFINITION 4.5: Let V = ; ∈≤≤
be a
super column vector space of refined labels over the field LR (or R); clearly V is an abelian group under addition. Wewillillustratethissituationbysomeexamples.
Example 4.14: LetV=
L L L L LL;1i8 L L L L
1 2 3 4 i 5 6 7 8
a a a a aR a a a a
∈≤≤ begroup
underaddition.Visasupercolumnvectorspaceofrefined labelsoverthefieldLR(orR).
L L LLL;1i5 L L
Example 4.15: LetV=
1 2 3i 4 5
a a aaR a a
∈≤≤ besuper columnvectorspaceofrefinedlabelsoverthefieldLR.
112
Example 4.16: LetV=
L L L LLL;1i7 L L L
1 2 3 4i 5 6 7
a a a aaR a a a
∈≤≤ beasuper
columnvectorspaceofrefinedlabelsoverthefieldLR(orR).
Example 4.17: LetV=
L L L LLL;1i7 L L L
1 2 3 4i 5 6 7
a a a aaR a a a
∈≤≤ beasuper
columnvectorspaceoverthefieldLR.
WecandefinethesupercolumnvectorsubspaceofV,Va supercolumnvectorspaceoverLR.
Wewillonlyillustratethesituationbysomeexamples.
113
Example
114
4.18: LetV= 1 2 3 4i 5 6 7 a a a aaR a a a L L L LLL;1i7 L L L ∈≤≤ beasuper
columnvectorspaceoverLR. LetW1= 1 2 i 3 4 a a aR a a L 0 L 0LL;1i4 L 0 L ∈≤≤⊆ Vbeasuper
columnvectorsubspaceofVoverLR. W2= 1 2i 3 a aaR a 0 0 L LLL;1i3 L 0 0 ∈≤≤⊆ Visasupercolumn vectorsubspaceofVoverLR.
115
W3= i 1 2 aR a a 0 0 0 0LL;1i2 0 L L ∈≤≤⊆ Visasupercolumn
vectorsubspaceofVofrefinedlabelsoverLR. Example 4.19: LetV= 1 2 3 4 5i 6 7 8 9 a a a a aaR a a a a L L L L LLL;1i9 L L L L ∈≤≤ beasuper columnvectorspaceofrefinedlabelsoverLR.
116
TakeW1=1i 2 3 4 aaR a a a 0 0 0 0 LLL;1i4 L L 0 L ∈≤≤⊆ V;isasuper columnvectorsubspaceofrefinedlabelsoverLR. W2= 1 2 i 3 a a aR a 0 L L 0 0LL;1i3 L 0 0 0 ∈≤≤⊆ Visasupercolumn vectorsubspaceofrefinedlabelsoverLR.
117
1 2 3 4 5
6 7 8 9 10 a a a a a
a a
L L L L L LLL;1i10 L L L L ∈≤≤
1 2 3 i 4 5 a a a aR a a L 0 L 0 L 0LL;1i5 L 0 0 L ∈≤≤⊆ V,isasuper
Example 4.20: LetV=
i
aR a a
a
bea supercolumnvectorspaceofrefinedlabelsoverLR. ConsiderW1=
columnvectorsubspaceofrefinedlabelsoverLR.
supercolumnvectorspaceofrefinedlabelsoverthefieldLR. Considerthefollowingsupercolumnvectorsubspacesof refinedlabelsoverLRofV.
118
0 L L L 0 LLL;1i7 L L L
TakeW2= 1 2 3 i 4 5 6 7 a a a aR a a a a
0 ∈≤≤⊆ Visasuper
11 12
L L L L L L LL;1i12 L L L L L L
columnvectorsubspaceofrefinedlabelsoverLR. Example 4.21: LetV= 1 2 3 4 5 6 i 7 8 9 10
a a a a a a aR a a a a a a
∈≤≤ bea
119
L L L 0 0 0 LL;1i7 L L L L 0 0 ∈≤≤⊆
0 0 0 L L L LL;1i4 0 0 0 0 L 0 ∈≤≤⊆ Visasupercolumn
W1= 1 2 3 i 4 5 6 7 a a a aR a a a a
Visasupercolumn vectorsubspaceofVoverLR. W2= 1 2 3 i 4 a a a aR a
vectorsubspaceofVoverLR.
120 W3=1 1 aR a 0 0 0 0 0 0 LL 0 0 0 0 0 L ∈⊆ Visasupercolumnvector
Wi ∩ Wj= 0 0 0 0 0 0 0 0 0 0 0 0 ifi ≠ j,1 ≤ i,j ≤ 3.
subspaceofrefinedlabelsofVoverLR.WeseeV= 3 1 i i W = and
121
1 2 3 i 4 5 6 a a a
a a a L L L LL;1i6 L L L ∈≤≤
columnvectorspaceofrefinedlabelsonLR.TakeW
1 2 i a a aR 0 L L LL;1i2 0 0 0 ∈≤≤⊆ V,W2= 1 i a aR L 0 0 LL 0 0 0 ∈⊆ V andW
i 1 2 3 aR a a a 0 0 0 LL;1i3 L L L ∈≤≤⊆ Vbethreesupercolumn vectorsubspaceofrefinedlabelsofVoverLR. WeseeV= 3 i i1 W = andWi ∩ Wj= 0 0 0 0 0 0 ifi ≠ j,1 ≤
≤ 3. ThusVisadirectunionofsupervectorcolumnsubspaces
R.
Example 4.22: LetV=
aR
beasuper
1=
3=
i,j
ofVoverL
122
1 2 3 4 i 5 6 7 8 a a a a aR a a a a L L L L LL;1i8 L L L L ∈≤≤
columnvectorspaceofrefinedlabelsoverLR(orR).Consider W1= 1 2 3 i 4 a a a aR a L L L 0 LL;1i4 0 0 L 0 ∈≤≤⊆ V,beasupercolumnvector subspaceofrefinedlabelsofVoverLR, W2= 1 2 i 3 4 a a aR a a 0 L 0 L LL;1i4 L L 0 0 ∈≤≤⊆ Vbeasupercolumn vectorsubspaceofrefinedlabelsofVoverLR.
Example 4.23: LetV=
beasuper
123
1
L 0 L 0
L 0 0 L
0 L
ClearlyV= 4 i
W = andWi ∩ Wj ≠ 0 0 0 0 0 0 0 0 ;i ≠ j,1 ≤
≤
W3=
2 i 3 4 a a aR a a
LL;1i4
∈≤≤⊆ Vbeasupercolumn vectorsubspaceofrefinedlabelsofVoverLR. W4= 1 2 i 3 4 5 6 a a aR a a a a
L 0 LL;1i6 L L L L ∈≤≤⊆ Vbeasupercolumn vectorsubspaceofrefinedlabelsoverLRofV.
i1
i
4. ThusVisapseudodirectsumofsupercolumnvector subspacesofVoverLR.
Itispertinenttomentionthatwedonothavesupercolumn linearalgebrasofVofrefinedlabelsoverLR.Howeverwecan alsohavesubfieldsupercolumnvectorsubspaceofrefined labelsoverLR.
Wewillillustratethisbysomeexamples.JustweknowLQ isafieldofrefinedlabelsandLQ ⊆ LR.Thuswecanhave subfieldsupercolumnvectorsubspacesaswellassubfield superrowvectorsubspacesofrefinedlabelsoverthesubfield LQofLR(orQofR).
columnvectorspaceofrefinedlabelsovertherefinedlabels fieldLQ.(LQ ⊆ LR).Misthusthesubfieldsupercolumnvector subspaceofrefinedlabelsoverthesubfieldLQofLR.
124
L L
Example 4.24: LetV= 1 2 3 4 i 5 6 7 8 a a a a aR a a a a L
L LLL;1i8 L L L ∈≤≤ beasuper
columnvectorspaceofrefinedlabelsoverthefieldLR(orR). TakeM= 1 2 i 3 a a aR a L 0 0 L 0LL;1i3 0 0 L ∈≤≤⊆ Vbeasuper
Example 4.25: LetM= (){ 12345678910 aaaaaaaaaa LLLLLLLLLL } i aR LL;1i10 ∈≤≤ beasuperrowvectorspaceofrefinedlabels overLR. TakeP= (){ 12345678910 aaaaaaaaaa LLLLLLLLLL } i aQ LL;1i10∈≤≤⊆ M;Pisasuperrowvectorspaceof refinedlabelsoverLQ;thusPisasubfieldsuperrowvector subspaceofVofrefinedlabelsoverthesubfieldLQofLR.We haveseveralsuchsubfieldvectorsubspacesofV.
Example 4.26: LetV=
∈≤≤ beasuper columnvectorspaceofrefinedlabelsoverthefieldLQ.Clearly Vhasnosubfieldsupercolumnvectorsubspacesofrefined labelsofVoverthesubfieldoverLQasLQhasnosubfields.In thiscasewecallsuchspacesaspseudosimplesupercolumn vectorsubspacesofrefinedlabelsofVoverLQ.HoweverV hasseveralsupercolumnvectorsubspacesofrefinedlabelsof
125
L L L L LLL;1i8 L L L
1 2 3 4 i 5 6 7 8 a a a a aQ a a a a
VoverLQ.ForinstancetakeK=
L 0 L 0 LLL;1i4 0 L 0
1 2 i 3 4
a a aR a a
∈≤≤⊆
V,KisasupercolumnvectorsubspaceofVoverthefieldLRof refinedlabels. Example 4.27: LetV= (){ 12345678910 aaaaaaaaaa LLLLLLLLLL } i aR LL;1i10 ∈≤≤ beasuperrowvectorspaceofrefined labelsoverthefieldLQ.
ClearlyLQhasnopropersubfieldsotherthanitselfasLQ ≅ Q.HenceVcannothavesubspaceswhicharesubfieldsuper rowvectorsubspaceofrefinedlabelsoverasubfieldofLQsoV isapseudosimplesuperrowvectorspaceofrefinedlabelsover LQ.
126
127
L L L L L LLL;1i10 L L L
L L 0 0 0LL;1i4 0 0 0 L
Example 4.28: LetV= 1 2 3 4 5 i 6 7 8 9 10 a a a a a aR a a a a a
L ∈≤≤ beasuper columnvectorspaceofrefinedlabelsoverLQ.Visapseudo simplesupercolumnvectorspaceofrefinedlabels.HoweverV hasseveralsupercolumnvectorsubspacesofrefinedlabelsover LQofV. ForinstanceT= 1 2 3 i 4 a a a aR a L
∈≤≤⊆ Visasuper columnvectorsubspaceofVofrefinedlabelsoverLQ. Nowwecanasincaseofothervectorspacesdefinebasis linearlyindependentelementlinearlydependentelementlinear
operatorandlineartransformation.Thistaskisleftasan exercisetothereader.Howeverallthesesituationswillbe describedbyappropriateexamples.
128
Example 4.29: LetV= 1 2 3 i 4 5 6 a a a aR a a a L L L LL;1i6 L L L ∈≤≤ beasuper columnvectorspaceofrefinedlabelsovertherefinedfieldLR. ConsiderthesetB= m1 m1 m1 m1 m1 L0000 0L000 00L00 ,,,, 000L0 0000L 00000 + + + + + ⊆ V.Itiseasily verifiedBisalinearlyindependentsubsetofVbutisnotabasis ofV.WeseeBcannotgenerateV. NowconsiderC= 1 1 1 2 a m1 am1 a a L00L0 0L00L 00000 ,,,,0000L 00000 000L0 + + ⊆ V;ItiseasilyverifieldthesubsetCofVcannotgenerateVsois notabasisfurtherCisnotalinearlyindependentsubsetofVas
RfromLQweseethedimension ofVoverLQisinfinite.
Thusasincaseofusualvectorspacesofdimensionofa supercolumnvectorspaceofrefinedlabelsdependsonthefield overwhichtheyaredefined.
Ritisof
129
0
wesee m1 L
0 0 0 0 + and 1a L 0 0 0 0 0 arelinearlydependentoneachother.
Thesearemorethanonepairofelementswhicharelinearly dependent. NowconsiderH= m1 m1 m1 m1 m1 m1 L00000 0L0000 00L000 ,,,,, 000L00 0000L0 00000L + + + + + + ⊆ Visabasis ofVandthespaceisoffinitedimensionanddimensionofV overLRissix. NowifweshiftthefieldL
ThisaboveexampleshowsifVisdefinedoverL
dimensionsixbutifVisdefinedoverLQthedimensionofV overLQisinfinite. Example 4.30: LetV= (){ 12345678910 aaaaaaaaaa LLLLLLLLLL } i aR LL;1i10 ∈≤≤ beasuperrowvectorspaceofrefined labelsovertherefinedfieldLQ.ClearlyVisofinfinite
130
()
()
()}
⊆ V,BisalinearlyindependentsubsetofVoverL
()
()
V,Pisa
VisofinfinitedimensionoverL
Example 4.31: LetV= () {} 12345i aaaaaaQ LLLLLLL;1i5 ∈≤≤ beasuperrow vectorspaceofrefinedlabelsoverthefieldLQ.ClearlyVis finitedimensionalanddimensionofVoverLQisgivenbyB= ()()(){ m1m1m1 L0000,0L000,00L00, +++ ()()} m1m1000L0,0000L++⊆ V,whichisthe basisofVoverLQ.
dimensionoverLQ.FurtherVhasalinearlyindependentas wellaslinearlydependentsubsetoverLQ. ConsiderB= (){ 1a L000000000,
2 a 00L0000000,
3 a 00000L0000, () 56 aa 000000L0L0
78i aaaR 0000000L0LLL;1i8 ∈≤≤
Q. TakeP= (){ 12 aa LL00000000,
12 aa 00LL000000,
22 aa 0L00L00000, () 114 aaa L0L00L0000, () 5 a 000000000L ()} 123 aaa 00LLL00000 ⊆
linearlydependentsubsetofVoverLQ.Wehaveseenthough
QstillVcanhavebothlinearly dependentsubsetaswellaslinearlyindependentsubset.
NowwewillgivealinearlyindependentsubsetofVwhich isnotabasisofV.
ConsiderT= ()()() {} 1 am1m1m1 L0000,00L00,000LL +++ inV;clearlyTisnotabasisbutTisalinearlyindependent subsetofVoverLQ.
Nowhavingseentheconceptofbasis,linearlyindependent subsetandlinearlydependentsubsetwenowproceedonto definethenotionoflineartransformationandlinearoperatorin caseoftherefinedspaceoflabelsoverLR(orLQ).
DEFINITION 4.6: Let V and W be two super row vector spaces of refined labels defined over the refined field LR (or LQ). Let T be a map from V into W. If T (cα + β) = cT (α) + T (β) for all c ∈ LR (or LQ) and α,β ∈ V then we define T to be a linear transformation from V into W.
131
ConsiderM= ()()(){ 112 am1aa L0000,L0000,0LL00, + ()()() 11 m1m1ab 0LL00,000L0,000L0 ++ } 1i baQ L,LL;1i2∈≤≤⊆ VisasubsetofVbutisclearlynota linearlyindependentsetonlyalinearlydependentsubsetofV.
() {} 123456i aaaaaaaR LLLLLLLL;1i6
(){ 12345678910 aaaaaaaaaa LLLLLLLLLL = } i aR LL;1i10 ∈≤≤ betwosuperrowvectorspacesofrefined
NoteifW=Vthenwedefinethelineartransformationto bealinearoperatoronV. Wewillillustratethesesituationsbysomeexamples. Example 4.32: LetV=
∈≤≤ andW
labelsoverLR.
132
→
T
1 2 3
4 5
L L L LL;1i6 L L L
1 2 3 4 5 i 6 7 8 9 10
a a a a a L L L L L LLL;1i10 L L L L ∈≤≤
DefineT:V
Wbesuchthat
() () 123456 aaaaaa LLLLLL = () 123456 aaaaaa LL000LLL0L;itiseasily verifiedTisalineartransformationofsuperrowvectorspaceof refinedlabelsoverLR. Itisstillinterestingtofindthenotionsofinvertiblelinear transformationandfindthealgebraicstructureenjoyedbythe collectionofalllineartransformationsfromVtoW. Example 4.33: IfV=
i
6 a a a aR a a a
∈≤≤ and W=
a a a a a aR
betwosupercolumn vectorspacesofrefinedlabelsoverLR.
133 WecandefinethemapT:V → Wby T( 1 2 3 4 5 6 a a a a a a L L L L L L )= () 123456 aaaaaa LLLLLL0000. ItiseasilyverifiedTisalineartransformationofVintoW. InfactweseekernelTisjustthezerosupervectorsubspaceof V. Example 4.34: LetV= 1 2 3 4 5i 6 7 8 9 a a a a aaR a a a a L L L L LLL;1i9 L L L L ∈≤≤ beasuper columnvectorspaceofrefinedlabelsoverLR.
DefineT:V →
itiseasilyverifiedthatTisalinearoperatoronVandkerTisa nontrivialsupervectorcolumnsubspaceofV.ClearlyTisnot onetoone.
Interestedreadercandefineonetoonemapsandstudythe algebraicstructureenjoyedbyR L Hom(V,V)={setofalllinear operatorsofVtoV}.Furtheritisinformedthatthereadercan derivealltheresultsandtheoremsprovedinthecaseofsuper rowvectorspaceswithappropriatemodificationsincaseof supercolumnvectorspaces.Thistaskisamatterofroutineand henceisleftforthereaderasanexercise.
Nowweproceedontodefinethenotionofsupervector spacesofm × nsupermatricesofrefinedlabelsoverthefield LR.
DEFINITION 4.7: Let V = {All m × n super matrices of refined labels with the same type of partition with entries from LR}; V is clearly an abelian group under addition. V is a super vector space of m × n matrices of refined labels over the field LR (or R or LQ or Q) .
Wewillillustratethissituationbysomesimpleexamples.
134
VbyT=( 1 2 3 4 5 6 7 8 9 a a a a a a a a a L L L L L L L L L )= 1 2 3 4 5 a a a a a L 0 0 L L L 0 0 L ;
Example
135
4.35: LetV= 123 456 i 789 101112 aaa aaa aR aaa aaa LLL LLL LL;1i12 LLL LLL ∈≤≤ bethesupermatrix vectorspaceofrefinedlabelsoverLR.
Example 4.36: LetM= 12 34 56 78 910i 1112 1314 1516 1718 aa aa aa aa aaaR aa aa aa aa LL LL LL LL LLLL;1i18 LL LL LL LL ∈≤≤ bea supermatrixvectorspaceofrefinedlabelsoverLR.Misalso knownasthesupercolumnvectorvectorspaceofrefinedlabels overLR.ThesesupermatricesinMarealsoknownasthesuper columnvectors(referchapterIofthisbook). Example 4.37: LetT= 147101316 258111417i 369121518 aaaaaa aaaaaaaR aaaaaa LLLLLL LLLLLLLL;1i18 LLLLLL ∈≤≤ bea supermatrixofvectorspaceofrefinedlabelsovertherefined fieldLR.WecanalsocallTtobeasuperrowvector–vector spaceofrefinedlabelsoverLR.
Example 4.38: LetK= 1234 5678 i 9101112 13141516
LLLL LLLL LL;1i16 LLLL LLLL
aaaa aaaa aR aaaa aaaa
∈≤≤ bethesuper matrixofvectorspaceofrefinedlabelsoverLR
Nowitispertinenttomentionherethatwecannotconstruct superlinearalgebraofrefinedlabelsevenifthenaturalorderof thesupermatrixisasquarematrix.Thuswefinditdifficultto obtainlinearalgebrasexpectthoseusingLRorLQoverLRorLQ respectively.
ThusweseeifV=LR= {} ii aaR LLL ∈ thenVisavector spaceaswellaslinearalgebraoverLRorLQhoweveritisnota superlinearalgebraofrefinedlabelsoverLRorLQ.
LikewiseLQ= {} ii aaQ LLL ∈ isonlyalinearalgebraover LQ,howeveritisnotsuperlinearalgebraoverLQ.
Nowwecandefinethenotionofsupervectorsubspaces, basis,linearoperatorandlineartransformationasincaseof superrowvectorspacesorsupercolumnvectorspaces.This taskisleftasanexercisetothereader.Weonlygiveexamples ofalltheseconceptssothatthereaderhasnodifficultyin followingthem.
136
Example 4.39: LetV=
LL LL LL LL LLLL;1i16 LL LL LL
12 34 56 78 i 910 1112 1314 1516
aa aa aa aa aR aa aa aa aa
∈≤≤ bea
supermatrixofvectorspaceofrefinedlabelsoverLR.(super columnvector,vectorspaceofrefinedlabels).
ConsiderM=
LL 00 LL 00 LLLL;1i8 00 LL 00
aa aa aR aa aa
12 34 i 56 78
∈≤≤⊆ V,isthe supercolumnvectorsubspaceofVoverLR.
137
138 TakeP= 12 34 i 56 aa aa aR aa 00 00 LL LL LLLL;1i6 00 00 00 ∈≤≤⊆ Visasuper columnvector,vectorsubspaceofVoverLRofV. Example 4.40: LetK= 123456 789101112 i 131415161718 192021222324 aaaaaa aaaaaa aR aaaaaa aaaaaa LLLLLL LLLLLL LL;1i24 LLLLLL LLLLLL ∈≤≤ be asuperrowvector,vectorspaceofrefinedlabelsoverLR. ConsiderM= 159 2610 i 3711 4812 aaa aaa aR aaa aaa 00L0LL 00L0LL LL;1i12 00L0LL 00L0LL ∈≤≤⊆ K,isa superrowvector,vectorsubspaceofKoverLR. Wecanhavemanysuchsuperrowvector,vectorsubspaces ofKoverR.
139
Example 4.41: LetV= 14 25i 36 aa aaaR aa LL LLLL;1i6 LL ∈≤≤ bea supermatrixvectorspaceofrefinedlabelsoverthefieldLR (orR). Example 4.42: LetP= 123 456 789 i 101112 131415 161718 aaa aaa aaa aR aaa aaa aaa LLL LLL LLL LL;1i18 LLL LLL LLL ∈≤≤ beasupermatrix vectorspaceofrefinedlabelsoverthefieldLR.TakeM= 1 2 34 i 5 67 8 a a aa aR a aa a 00L 00L LL0 LL;1i8 00L LL0 00L ∈≤≤⊆ Pisasupermatrix vectorsubspaceofPoverLR. Example 4.43: LetV= 123 456i 789 aaa aaaaR aaa LLL LLLLL;1i9 LLL ∈≤≤ beasupermatrixvectorspaceofrefinedlabelsoverLR
RofV.
nmatrixofrefinedlabel
subspacesandpseudodirectsumofsupervectorsubspaces.
140
TakeM= 1 2i 34 a aaR aa 00L 00LLL;1i4 LL0 ∈≤≤⊆ V,bea
3
L00 LL;1i3 0L0
3 i 1 2
aR a a 0L0 000 LL;1i3 L00 L00 ∈≤≤⊆ V,
supermatrixvectorsubspaceofrefinedlabelsoverL
Nowasincaseofsuperrowvectorspaceofrefinedlabels wecaninthecaseofgeneralsuperm ×
ofvectorspacedefinethenotionofdirectsumofvector
Thistaskissimpleandhenceisleftasanexercisetothereader. Howeverweillustratethissituationbysomeexamples. Example 4.44: LetV= 123 456 i 789 101112 aaa aaa aR aaa aaa LLL LLL LL;1i12 LLL LLL ∈≤≤ beasupermatrix vectorspaceofrefinedlabelsoverthefieldLR.ConsiderW1= 1 2 i
a a aR a L00
000 ∈≤≤⊆ V,beasupermatrixrow vectorsubspaceofVoverLR. W2=
a
141 W3= 1 2 i a a aR 00L 00L LL;1i2 000 000 ∈≤≤⊆ V, W4= 1 i 2 a aR a 000 0L0 LL;1i2 00L 000 ∈≤≤⊆ Vand W5= i 12 aR aa 000 000 LL;1i2 000 0LL ∈≤≤⊆ V,besupermatrix vectorsubspacesofVoverLR.WeseeclearlyV= 5 i i1 W = and Wi ∩ Wj= 000 000 000 000 ifi ≠ j,1 ≤ i,j ≤ 5.ThusVisthedirect sumofsupermatrixvectorsubspacesofVoverLR. Example 4.45: LetV= 1234567 891011121314i 15161718192021 aaaaaaa aaaaaaaaR aaaaaaa LLLLLLL LLLLLLLLL;1i21 LLLLLLL ∈≤≤ beasuperrowvector,vectorspaceofrefinedlabelsoverLR.
Consider
142
superrowvector,vectorsubspacesofVoverthefieldLR
W1= 1 2i 3 a aaR a L000000 L000000LL;1i3 L000000 ∈≤≤⊆ V, W2= 1 2i 3 a aaR a 0L00000 0L00000LL;1i3 0L00000 ∈≤≤⊆ V, W3= 12 i 34 aa aR aa 00LL000 0000000LL;1i4 00LL000 ∈≤≤⊆ V, W4= 3 124i 5 a aaaaR a 0000L00 00LLL00LL;1i5 0000L00 ∈≤≤⊆ V, W5= 1 2i 3 a aaR a 00000L0 00000L0LL;1i3 00000L0 ∈≤≤⊆ Vand W6= 1 2i 3 a aaR a 000000L 000000LLL;1i3 000000L ∈≤≤⊆ Vbe
.
143 ClearlyV= 6 i i1 W = andWi ∩ Wj= 0000000 0000000 0000000 ; ifi ≠ j.ThusweseeVisadirect sumofsuperrowvectorsubspaceofVoverLR. Example 4.46: LetV= 1234 5678 i 9101112 13141516 aaaa aaaa aR aaaa aaaa LLLL LLLL LLLLLL;1i16 LLLL ∈≤≤ beasuper matrixvectorspaceofrefinedlabelsoverLR. ConsiderW1= 1 2 i 3 a a aR a L000 L000 L000LL;1i3 0000 ∈≤≤⊆ V, W2= i 2 134 aR a aaa 0000 0000 0L00LL;1i4 LL0L ∈≤≤⊆ V, W3= 2 3 i 4 1 a a aR a a 000L 000L 000LLL;1i4 00L0 ∈≤≤⊆ V;
144 W4= 1 2 i 3 a a aR a 0L00 0L00 00L0LL;1i3 0000 ∈≤≤⊆ Vand W5= 1 2 i a a aR 00L0 00L0 0000LL;1i3 0000 ∈≤≤⊆ V besupermatrixvectorsubspacesofVovertherefinedfieldLR. ClearlyV= 5 i i1 W = andWi ∩ Wj= 0000 0000 0000 0000 ifi ≠ j, 1 ≤ i,j ≤ 5.ThusVisthedirectsumofsupervectorsubspaces ofVoverLR. Nowhavingseentheconceptofdirectsumwenowproceed ontogiveexamplesofpseudodirectsumofsubspaces. Example 4.47: LetV= 12345 678910i 1112131415 aaaaa aaaaaaR aaaaa LLLLL LLLLLLL;1i15 LLLLL ∈≤≤ beasuper rowvector,vectorspaceofrefinedlabelsoverLR.
145
a
2
14 a
∈≤≤⊆
6
W
23 456i 1789
00LLLLL;1i9 L0LLL ∈≤≤⊆ Vare superrowvectorsubspacesofVoverLR. WeseeclearlyV= 5 i i1 W = andWi ∩ Wj ≠ 00000 00000 00000 ifi ≠ j,1 ≤ i,j ≤ 5. ThusVisnotthedirectsumofsuperrowvectorsubspaces ofVbutonlyapseudodirectsumofsuperrowvector subspacesofVoverLR.
Consider W1= 1 2i 345
aaR aaa L0000 L0000LL;1i5 L0L0L ∈≤≤⊆ V, W2= 3i 124 aaR aaa 00000 00L00LL;1i4 LL00L ∈≤≤⊆ V, W3=
3i
aaR aa 0L000 0L000LL;1i4 LL000
V, W4= 2345 1i
aaaa aaR a 0LLLL L0000LL;1i6 0000L ∈≤≤⊆ Vand
5=
aa aaaaR aaaa 0LL00
146
Example 4.49:LetV= 123 456 789 101112 131415i 161718 192021 222324 252627 aaa aaa aaa aaa aaaaR aaa aaa aaa aaa LLL LLL LLL LLL LLLLL;1i27 LLL LLL LLL LLL ∈≤≤ beasupercolumnvector,vectorspaceofrefinedlabelsoverLR. ConsiderW1= 123 456 i 789 aaa aaa aR aaa LLL 000 LLL 000 000LL;1i9 LLL 000 000 000 ∈≤≤⊆ V,
W2=
L00 000 0L0 000 000LL;1i9 00L LLL 000 LLL
W3=
a a aR a aaa aaa
1 2 i 3 456 789
∈≤≤⊆ V,
L00 LLL 0L0 LLL 000LL;1i8 000 000 000 000
1 234 5 678 i
a aaa a aaa aR
∈≤≤⊆ V,
147
148 W4= 1 2 3 456i 7 a a a aaaaR a L00 00L 000 00L LLLLL;1i7 000 000 000 00L ∈≤≤⊆ Vand W5= 1 2 3 4 5i 6 7 8910 11 a a a a aaR a a aaa a L00 00L L00 0L0 00LLL;1i11 0L0 L00 LLL L00 ∈≤≤⊆ Varesuper columnvector,vectorsubspacesofVoverLR.
149 ClearlyV= 5 i i1 W = andWi ∩ Wj ≠ 000 000 000 000 000 000 000 000 000 ifi ≠ j,1 ≤ i, j ≤ 5. ThusVisonlyapseudodirectsumofsupercolumnvector, vectorsubspacesofVoverLR. Example 4.49: LetV= 1234 5678 9101112 i 13141516 17181920 21222324 aaaa aaaa aaaa aR aaaa aaaa aaaa LLLL LLLL LLLL LLLLLL;1i24 LLLL LLLL ∈≤≤ beasuper matrixvectorspaceofrefinedlabelsoverLR.ConsiderW1= 13 24 i aa aa aR L00L L00L 0000 0000LL;1i4 0000 0000 ∈≤≤⊆ V,
150
W2= 123 456 i aaa aaa aR LLL0 0LLL 0000 0000LL;1i6 0000 0000 ∈≤≤⊆ V, W3= 24 13 i 567 aa aa aR aaa 0L0L L0L0 0000 0000LL;1i7 0000 L0LL ∈≤≤⊆ V, W4= 123 47 i 58 69 aaa aa aR aa aa LL0L 0000 L00L L00LLL;1i9 L00L 0000 ∈≤≤⊆ V, W5= 1 910 24 i 5 6 378 a aa aa aR a a aaa L000 00LL LL00 0L00LL;1i10 0L00 L0LL ∈≤≤⊆ V,and
151 W6= 1 4 6 i 27 89 53 a a a aR aa aa aa L000 00L0 00L0 L0L0LL;1i9 00LL LL00 ∈≤≤⊆ V,besuper vectorsubspaceofVoverLR.V= 6 1 i i W = andWi ∩ Wj ≠ 0000 0000 0000 0000 0000 0000 ifi ≠ j,1 ≤
.
Example
1234 5678 i 9101112 13141516 aaaa aaaa aR aaaa aaaa LLLL LLLL LL;1i16 LLLL LLLL ∈≤≤ beasupermatrix vectorspaceofrefinedlabelsoverL
.
i,j ≤ 6. ThusVisapseudodirectsumofsupermatrixvector subspacesofVoverLR
Nowwecanasincaseofvectorspacesdefinelinear transformation,linearoperatorandprojection.Thistaskisleft asanexercisetothereader.Alsoallsupermatrixvectorspaces ofrefinedlabelsoverLR.
4.50: LetVandWbetwosupermatrixvectorspaces ofrefinedlabelsoverLR,whereV=
R
152
ConsiderW= 12345 678910i 1112131415 aaaaa aaaaaaR aaaaa LLLLL LLLLLLL;1i15 LLLLL ∈≤≤ beasupermatrixvectorspaceofrefinedlabelsofLR. DefineT:V → Wby T 1234 12345 5678 678910 9101112 1112131415 13141516 aaaa aaaaa aaaa aaaaa aaaa aaaaa aaaa LLLL LLLLL LLLL LLLLL LLLL LLLLL LLLL = ClearlyTisalineartransformationofsupermatrixvector spaceofrefinedlabelsoverLR Example 4.51: LetV= 123 456 789i 101112 131415 aaa aaa aaaaR aaa aaa LLL LLL LLLLL;1i15 LLL LLL ∈≤≤ beasupermatrix vectorspaceofrefinedlabelsoverLR. ConsiderT:V → Vamapdefinedby T 123 456 789 101112 131415 aaa aaa aaa aaa aaa LLL LLL LLL LLL LLL = 12 3 46 5 78 aa a aa a aa LL0 00L 0LL L00 0LL
ClearlyTisalinearoperatoronV.
NowconsiderW=
LLL LLL LLL LLL LLL
aaa aaa aaa aaa aaa
123 456 789 101112 131415
=
123 456 789i
aaa aaa aaaaR
∈≤≤
LLL LLL LLLLL;1i9 000 000
⊆ V,WisasupermatrixvectorsubspaceofVofrefinedlabels overLR. P
aaa aaa aaa
LLL LLL LLL 000 000
123 456 789
.
ItiseasilyverifiedthatPisaprojectionofVtoWandP (W) ⊆ W;withP2=PonV.
Wecanasincaseofsuperrowvectorspacederiveallthe relatedtheoremsgiveninthischapterwithappropriate modificationsandthisworkcanbecarriedoutbythereader.
Howeverwerepeatedlymakeamentionthatsupermatrix linearalgebraofrefinedlabelscannotbeconstructedasitisnot possibletogetanycollectionofmultiplicativewisecompatible supermatricesoflabels.
Nowweproceedontodefinethenotionoflinearfunctional ofrefinedlabelsLR.
HereonceagainwecallthattherefinedlabelsLRis isomorphicwithR.Sowecantaketherefinedlabelsitselfasa fieldorRasafield.
153
LetVbeasupermatrixvectorspaceofrefinedlabelsover LR.DefinealineartransformationfromVintothefieldof refinedlabelscalledthelinearfunctionalofrefinedlabelsonV.
Iff:V → LRthen f(Lc α + β )=Lcf(α)+f(β )where α,β∈ VandLc ∈ LR. Wewillfirstillustratethisconceptbysomeexamples.
154
Example 4.52: LetV= 1 2 3 4 i 5 6 7 8 a a a a aR a a a a L L L L LL;1i8 L L L L ∈≤≤ beasuper columnvectorspaceofrefinedlabelsoverthefieldofrefined labelsLR. T=( 1 2 3 4 5 6 7 8 a a a a a a a a L L L L L L L L )=18 aa L...L ++ = 128 aa...a L +++∈ LR. TisalinearfunctionalonV.
Infacttheconceptoflinearfunctionalcanleadtoseveral interestingapplications.
155
L
L
Example 4.53: LetV= 1 2 3 4i 5 6 7 a a a aaR a a a L L L LLL;1i7 L L L ∈≤≤ beasuper columnvectorspaceofrefinedlabelsoverLR. Definef:V → LRby f( 1 2 3 4 5 6 7 a a a a a a a L L L
L
L )=123 aaa LLL ++ = 123aaa L ++∈ LR.fisalinearfunctionalonV. Considerf1:V → LRby
156 f1( 1 2 3 4 5 6 7 a a a a a a a L L L L L L L )=4 a L ;f1isalsoalinearfunctionalonV.We candefineseverallinearfunctionalonV. Considerf2( 1 2 3 4 5 6 7 a a a a a a a L L L L L L L )=147 aaa LLL ++ = 147aaa L ++∈ LR; f2isalinearfunctionalonV.Weseeallthethreelinear functionalsf1,f2andfaredistinct. Example 4.54: LetV= () {} 12345678i aaaaaaaaaR LLLLLLLLLL;1i8 ∈≤≤ beasuperrowvectorspaceofrefinedlabelsoverLR. Definef:V → LRby f( () 12345678 aaaaaaaa LLLLLLLL) = 1247 aaaa LLLL +++ = 1247 aaaa L +++ ;fisalinearfunctionalonV. WecandefineseverallinearfunctionalonV.
157
=
=
L
Example 4.55: LetV= 123 456 789 101112 131415 i 161718 192021 222324 252627 282930 aaa aaa aaa aaa aaa aR aaa aaa aaa aaa aaa LLL LLL LLL LLL LLL LL;1i30 LLL LLL LLL LLL LLL ∈≤≤ beasupercolumn vector,vectorspaceofrefinedlabelsoverLR,thefieldof refinedlabels. Definef:V → LRby f( 123 456 789 101112 131415 161718 192021 222324 252627 282930 aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa LLL LLL LLL LLL LLL LLL LLL LLL LLL LLL )
157111317192329 aaaaaaaaa LLLLLLLLL ++++++++
157111317192329 aaaaaaaaa
++++++++ ,fisalinearfunctionalonV.
158
Example 4.56: LetV= 159131721 2610141822 i 3711151923 4812162024 aaaaaa aaaaaa aR aaaaaa aaaaaa LLLLLL LLLLLL LL;1i24 LLLLLL LLLLLL ∈≤≤ bea superrowvectorspaceofrefinedlabelsoverLR Considerf:V → LRby f( 159131721 2610141822 3711151923 4812162024 aaaaaa aaaaaa aaaaaa aaaaaa LLLLLL LLLLLL LLLLLL LLLLLL )=161124 aaaa LLLL +++ = 161124 aaaa L +++ isalinearfunctionalonV.
Example 4.57: LetV= 123 456 789i 101112 131415 aaa aaa aaaaR aaa aaa LLL LLL LLLLL;1i15 LLL LLL ∈≤≤ beasupermatrix supervectorspaceofrefinedlabelsoverLR,therefinedfieldof labels. Definef:V → LRby f( 123 456 789 101112 131415 aaa aaa aaa aaa aaa LLL LLL LLL LLL LLL )
= 1571161213 aaaaaaa LLLLLLL ++++++ = 1571161213 aaaaaaa L ++++++ ; fisalinearfunctionalonV.
Weasincaseofvectorspacedefinethecollectionofall linearfunctionalsonV,VasupermatrixvectorspaceoverLR asthedualspaceanditisdenotedbyL(V,LR).
Herewedefine ij L δ =0ifi ≠ jifi=jthenij L δ =Lm+1.
Usingthisconventionwecanproveallresultsanalogousto L(V,F),Fanyfield.
ForinstanceifV= () {} 1234i aaaaaR LLLLLL;1i4 ∈≤≤ beasuperrow vectorspaceofrefinedlabelsoverLRthenB={(Lm+1|0|00) =v1,v2=(0|Lm+1|00),v3=(0|0|Lm+1|0),v4=(0|0|0 Lm+1)} ⊆ VisabasisofVoverLR.
Definefi(vj)=ij L δ ;1 ≤ i,j ≤ 4;(f1,f2,f3,f4)isabasisof L(V,LR)overLR.
Weasincaseofusualvectorspacesdefineiff:V → LRis alinearfunctionalonsupervectorspaceofrefinedlabelswith appropriatemodificationsthennullspaceoffisofdimension n–1.
Itishoweverpertinenttomentionherethatwemayhave severalsupervectorspacesofdimensionnwhichneednotbe isomorphicasincaseofusualvectorspaceswiththisinmind wecansayorderiveresultswithsuitablemodifications.
HoweverforanysupermatrixvectorspaceVofrefined labelsoverLRtherefinedhypersuperspaceofVasasuper subspaceofdimensionn–1wherenisthedimensionofV.
159
ConsiderV=12 i 34
LL LL;1i4 LL ∈≤≤ beasuper
aa aR aa
matrixvectorspaceofdimensionfourovertherefinedfieldLR.
ConsiderW=12 i 3
aa aR a
LL LL;1i3 0L ∈≤≤⊆ V,Wis thehypersuperspaceofVanddimensionofWisthreeoverLR.
Example 4.58: LetV=
L L L L L LLL;1i10 L L L L
1 2 3 4 5 i 6 7 8 9 10
a a a a a aR a a a a a
∈≤≤ beasuper
columnvectorspaceofdimension10overthefieldLRof refinedlabels.
160
161 ConsiderM= 1 2 3 4 i 5 6 7 8 9 a a a a aR a a a a a L L L L 0 LLL;1i9 L L L L ∈≤≤⊆ Visahyper superspaceofrefinedlabelsofVoverLR.InfactVhasseveral hypersuperspaces. Example 4.59: LetV= 123 456 789 i 101112 131415 161718 aaa aaa aaa aR aaa aaa aaa LLL LLL LLL LL;1i18 LLL LLL LLL ∈≤≤ beasupermatrix vectorspaceofrefinedlabelsoverLR. ConsiderW= 12 345 678 i 91011 121314 151617 aa aaa aaa aR aaa aaa aaa 0LL LLL LLL LL;1i17 LLL LLL LLL ∈≤≤⊆ V, beasupermatrixvectorsubspaceofVoverLR.
ClearlyWisahypersuperspaceofrefinedlabelsoverLR ofV.DimensionofVis18whereasdimensionofWoverLR is17.
LetVbeasupervectorspaceofrefinedlabelsoverLR.S beasubsetofV,theannihilatorofSisthesetS oofallsuper linearfunctionalsfonVsuchthatf(α)=0forevery α ∈ S. LetV= {} 123i aaaaR LLLLL;1i3 ∈≤≤ beasuper rowvectorspaceofrefinedlabelsoverLR.Wewillonlymake atrial. SupposeW= {} 1i aaR L00LL∈⊆ Vbeasubsetof V. Definef:V → Vby f( () 1a L00)= () 000every () 1a L00 ∈ W. Wo={f ∈ L(V,LR)|f( () 1a L00) = () 000forevery1 a L ∈ LR}. WeseeWoisasubspaceofL(V,LR)asitcontainsonlyf andfothezerofunction.Howeveronehastostudyinthis directiontogetanalogousresults. Wewillgivesomemoreexamples. ConsiderK= 12345 678910 i 1112131415 1617181920
aaaaa aaaaa
162
aR aaaaa aaaaa LLLLL LLLLL LL;1i20 LLLLL LLLLL ∈≤≤ beasupermatrixvectorspaceofrefinedlabelsoverthe fieldLR.
163
LetS= 1 2 a a 00000 00000 0L000 L0000 , 12 34 aa aa LL000 LL000 00000 00000 , 123 aaa 00LLL 00000 00000 00000 where i a L ∈ LR;1 ≤ i ≤ 4} ⊆ Vbea propersubsetofV.ClearlySisnotasupervectorsubspaceof VitisonlyapropersubsetofV. Definefi:V → Vasfollows: f1( 12345 678910 1112131415 1617181920 aaaaa aaaaa aaaaa aaaaa LLLLL LLLLL LLLLL LLLLL ) = 121314 5151617 8181920 aaa aaaa aaaa 00000 00LLL L0LLL 0LLLL , f2( 12345 678910 1112131415 1617181920 aaaaa aaaaa aaaaa aaaaa LLLLL LLLLL LLLLL LLLLL )=8 a 00000 00L00 00000 00000 ,
f3(
LLLLL LLLLL LLLLL LLLLL
aaaaa aaaaa aaaaa aaaaa
12345 678910 1112131415 1617181920
) = 1819 aa
00000 00LL0 00000 00000
, f4(
LLLLL LLLLL LLLLL LLLLL
aaaaa aaaaa aaaaa aaaaa
12345 678910 1112131415 1617181920
00000 00000 00LLL 00LLL
)= 123 456
andsoon. Weseef1,f2,f3andf4aredefinedsuchthatfi(x)= 00000 00000 00000 00000
aaa aaa
foreveryx ∈ S,1 ≤ i ≤ 4.Nowhavingseen howSolookslikewecangiveasubspacestructureofL(V,LR). Wecanasincaseofusualvectorspacederiveproperties relatedwiththedualsuperspaceandpropertiesenjoyedbythe superlinearfunctionalsorlinearfunctionalsonsupermatrix vectorspacesofrefinedlabels.TheonlycriteriabeingLRis isomorphictoR.Alsowehavethepropertiesusedinthesuper linearalgebraregardingsupervectorspaces[47].
164
Inthischapterweforthefirsttimeintroducethenotionof supersemivectorspaceofrefinedlabels.
Wehavedefinedthenotionofsemigroupofrefinedlabels. AlsowehaveseenthesetoflabelsLa ∈ R{0} L + ∪ formsa semifieldofrefinedlabels.Also Q{0} L + ∪ isagainasemifieldof refinedlabels.Wewouldbeusingthesemifieldofrefinedlabels tobuildsupersemivectorspacesofdifferenttypes.
ClearlyMisasemigroupunderaddition.WeseeMis infactonlyaninfinitecommutativesemigroupunderthe operationofaddition.ClearlyMisnotagroupunderaddition. FurtherwecannotdefineproductonMasitisnotcompatible
165
Let M= (){ 123456n1n aaaaaaaa LLLLLL...LL } i aR{0} LL;1in + ∪ ∈≤≤ bethecollectionofallsuperrow
vectorsofrefinedlabels.
underanyformofproductbeingasuperrowvector.Weknow R{0} L + ∪ and Q{0} L + ∪ aresemifields.Nowwecandefinesuper semivectorspaceofrefinedlabels.
DEFINITION 5.1: Let (){ 123456n1ni aaaaaaaaa R{0} VLLLLLL...LLLL; + ∪ =∈ 1 ≤ i ≤ n} be a semigroup under addition with () 000000000 LLLLLLLLL as the zero row vector of refined labels. V is a super semivector space of refined labels over + ∪ R{0} L ; the semifield of refined labels isomorphic with R+ ∪ {0}. Wewillfirstillustratethissituationbysomesimpleexamples.
Example 5.1: Let V= () {} 1234i aaaaaR{0} LLLLLL;1i4 + ∪ ∈≤≤ beasupersemivectorspaceofrefinedlabelsoverthesemifield R{0} L + ∪ . Example 5.2: Let V= (){ 12345678910 aaaaaaaaaa LLLLLLLLLL } i aR{0} LL;1i10 + ∪ ∈≤≤ beasupersemivectorspaceofrefined labelsoverthesemifield R{0} L + ∪ .WealsocallVtobeasuper rowsemivectorspaceofrefinedlabels.Sincefromthevery contextonecaneasilyunderstandwedonotusuallyqualify thesespacesbyroworcolumn.
Example 5.3: Let () {} 1234567i aaaaaaaaR{0} SLLLLLLLLL;1i7 + ∪ =∈≤≤ beasupersemivectorspaceofrefinedlabelsoverthesemifield R{0} L + ∪ .
166
167 Example 5.4: Let M= () {} 123456i aaaaaaaR{0} LLLLLLLL;1i6 + ∪ ∈≤≤ beasupersemivectorspaceofrefinedlabelsoverthesemifield Q{0} L + ∪ . Example 5.5: Let V= (){ 123456789 aaaaaaaaa LLLLLLLLL } i aR{0} LL;1i9 + ∪ ∈≤≤ beasupersemivectorspaceofrefined labelsover Q{0} L + ∪ thesemifieldofrefinedlabelsisomorphic withQ+ ∪ {0}. Example 5.6: Let () {} 12345678i aaaaaaaaaQ{0} KLLLLLLLLLL;1i8 + ∪ =∈≤≤ beasupersemivectorspaceofrefinedlabelsoverQ{0} L + ∪ . ClearlyKisnotasupersemivectorspaceofrefinedlabelsover R{0} L + ∪ . Example 5.7: Let V= (){ 12345678910 aaaaaaaaaa LLLLLLLLLL } i aQ{0} LL;1i10 + ∪ ∈≤≤ beasupersemivectorspaceofrefined labelsover Q{0} L + ∪ .Itispertinenttomentionherethatitdoes notmakeanydifferencewhetherwedefinethesupersemivector spaceoverR+ ∪ {0}orR{0} L + ∪ (orQ+ ∪ {0}orQ{0} L + ∪ )as R+ ∪ {0}isisomorphictoR{0} L + ∪ (Q+ ∪ {0}isisomorphicwith Q{0} L + ∪ ). Wecanasincaseofsemivectorspacesdefinesubstructures onthem.Thisissimpleandsoleftasanexercisetothereader. Wegivesomesimpleexamplesofsubstructuresandillustrate theirpropertiesalsowithexamples.
Example 5.8: Let V= (){ 123456789 aaaaaaaaa LLLLLLLLL } i aR{0} LL;1i9 + ∪ ∈≤≤ beasupersemivectorspaceofrefined labelsover R{0} L + ∪ .Consider M= (){ 12345 aaaaa LLL00L0L0 } i aR{0} LL;1i5 + ∪ ∈≤≤⊆ V,Misasupersemivectorsubspace ofVoverthesemifield R{0} L + ∪ .
Example 5.9: Let V= (){ 1234567891011 aaaaaaaaaaa LLLLLLLLLLL } i aR{0} LL;1i11 + ∪ ∈≤≤ beasupersemivectorspaceofrefined labelsoverthesemifield R{0} L + ∪ .Consider W= (){ 123456 aaaaaa LLLLLL00000 } i aR{0} LL;1i6 + ∪ ∈≤≤⊆ V,isasupersemivectorsubspaceof Vovertherefinedlabelsemifield R{0} L + ∪ .
Example 5.10: Let V= (){ 123456789101112 aaaaaaaaaaaa LLLLLLLLLLLL } i aR{0} LL;1i12 + ∪ ∈≤≤ beasupersemivectorspaceofrefined labelsoverthesemifield Q{0} L + ∪ . Consider (){ 123456789101112 aaaaaaaaaaaa WLLLLLLLLLLLL = } i aQ{0} LL;1i12 + ∪ ∈≤≤⊆ V;Wisasupersemivectorsubspace ofVofrefinedlabelsoverthesemifield Q{0} L + ∪ .
168
Example 5.11: Let
169
V= () {}
+ ∪
L + ∪ . Consider () {} 123i
+ ∪
⊆ V,isasupersemivectorsubspaceofVofrefinedlabelsover
L + ∪ .
V= () {} 123456i
LLLLLLLL;1i6 + ∪ ∈≤≤ beasupersemivectorspaceofrefinedlabelsoverthesemifield
L + ∪ . Consider M1= () {} 12i aaaR{0} L0L000LL;1i2 + ∪ ∈≤≤⊆ V, M2= () {} 12i aaaR{0} 0L000LLL;1i2 + ∪ ∈≤≤⊆ V, and M3= () {} 12i aaaR{0} 000LL0LL;1i2 + ∪ ∈≤≤⊆ V bethreesupersemivectorsubspaceofVofrefinedlabelsover R{0} L + ∪ .ClearlyV= 3 i i1 M = withMi ∩ Mj={(000|00|0)}if i ≠ j,1 ≤ i,j ≤ 3.ThusVisadirectsumsupersemivector subspacesM1,M2andM3ofV.
123456i aaaaaaaQ{0} LLLLLLLL;1i6
∈≤≤ beasupersemivectorspaceofrefinedlabelsoverthesemifield Q{0}
aaaaQ{0} ML0L0L0LL;1i3
=∈≤≤
Q{0}
Nowhavingseensupersemivectorsubspacesofasuper semivectorspacewecannowproceedontogiveexamplesof thenotionofdirectsumandpseudodirectsumofsuper semivectorsubspacesofasupersemivectorspace. Example 5.12: Let
aaaaaaaR{0}
ofrefinedlabels R{0}
170 Example 5.13: Let V= (){ 12345678910 aaaaaaaaaa LLLLLLLLLL } i aQ{0} LL;1i10 + ∪ ∈≤≤ beasupersemivectorspaceofrefined labelsover Q{0} L + ∪ . Take W1= () {} 12i aaaQ{0} L000L00000LL;1i2 + ∪ ∈≤≤⊆ V, W2= () {} 123i aaaaQ{0} 00000LLL00LL;1i3 + ∪ ∈≤≤⊆ V, W3= () {} 123i aaaaQ{0} 0LLL000000LL;1i3 + ∪ ∈≤≤⊆ V and W4= () {} 12i aaaQ{0} 00000000LLLL;1i2 + ∪ ∈≤≤⊆ V; besupersemivectorsubspacesofVovertherefinedlabel semifield Q{0} L + ∪ .WehaveV= 4 i i1 W = withWi ∩ Wj={(000 0|0|00|000)}ifi ≠ j,1 ≤ i,j ≤ 4.ThusVisadirectsum supersemivectorsubspacesW1,W2,W3andW4. Example 5.14: Let V= ({ 123456 aaaaaa LLLLLL ) 7891011121314 aaaaaaaa LLLLLLLL } i aR{0} LL;1i14 + ∪ ∈≤≤ beasupersemivectorspaceofrefined labelsoverthesemifieldofrefinedlabels R{0} L + ∪ . Consider (){ 12345 1aaaaa WL00LLL0000000L = } i aR{0} LL;1i5 + ∪ ∈≤≤⊆ V, W2= (){ 123456 aaaaaa 0LLL0L00LL0000
171 } i aR{0} LL;1i6 + ∪ ∈≤≤⊆ V, W3= (){ 1234576 aaaaaaa L000LLL00L00LL } i aR{0} LL;1i7 + ∪ ∈≤≤⊆ V, W4= (){ 12345677 aaaaaaaa LL0000LLLLL0L0 } i aR{0} LL;1i8 + ∪ ∈≤≤⊆ V and W5= (){ 12345678 aaaaaaaa 0L0L0LL0L0LLL0 } i aR{0} LL;1i8 + ∪ ∈≤≤⊆ V besupersemivectorsubspacesofVofrefinedlabelsover R{0} L + ∪ .WeseeV= 5 i i1 W = butWi ∩ Wj ≠ {(0|00|000|00 00|00|00)}ifi ≠ j,1 ≤ i,j ≤ 5.ThusweseeVisonlya pseudodirectsumofsupersemivectorsubspacesofVoverLR andisnotadirectsumofsupersemivectorsubspacesofVover LR. Example 5.15: Let () {} 12345678i aaaaaaaaaQ{0} VLLLLLLLLLL;1i8 + ∪ =∈≤≤ beasupersemivectorspaceofrefinedlabelsovertherefined labelssemifield Q{0} L + ∪ . Take P1= () {} 123i aaaaQ{0} 000LL00LLL;1i3 + ∪ ∈≤≤⊆ V, P2= () {} 1234i aaaaaQ{0} L0LL0L00LL;1i4 + ∪ ∈≤≤⊆ V, P3= () {} 12534i aaaaaaQ{0} L0LL0LL0LL;1i5 + ∪ ∈≤≤⊆ V, P4= () {} 125346i aaaaaaaQ{0} 0LLLL0LLLL;1i6 + ∪ ∈≤≤
andPi ∩ Pj ≠ {(0|00|0|0|0|0 0|0)},ifi ≠ j,1 ≤ i,j ≤ 8.ThusVisonlyapseudodirectsum ofsupersemivectorsubspacesP1,P2,…,P5ofVoverQ{0} L + ∪ . Nowitmaysohappensometimesforanygivensuper semivectorspaceVofrefinedlabelsoverthesemifieldQ{0} L + ∪ or R{0} L + ∪ thegivensetofsupersemivectorsubspacesofV maynotbesayasV ≠ n i i1 P = only n i i1 P = ⊄ V.Thissituationcan occurwhenwearesuppliedwiththesemivectorsubspacesof refinedlabels.Butifweconsiderthesupersemivector subspaceofrefinedlabelsweseethatwecanhaveeitherthe directunionorthepseudodirectunion.Nowwediscussabout thebasis.Sincewedonothavenegativeelementsina semifieldweneedtoapplysomemodificationsinthisregard. LetV= () {} 1234ni aaaaaaR{0} LLLL...LLL;1in + ∪ ∈≤≤ beasupervectorsemivectorspaceofrefinedlabelsover R{0} L + ∪ .WeseeB={[Lm+10|00|…|0],[0Lm+1|00|…|0], …,[00|00|…|Lm+1]}actsasabasisofVoverR{0}
172
and P5= () {} 1234i aaaaaQ{0} 0L0L0L0LLL;1i4 + ∪ ∈≤≤⊆ V besupersemivectorsubspacesofVofrefinedlabelsover Q{0} L + ∪ .WeseeV= 5 i i1 P = L + ∪ .We definethenotionoflinearlydependentandindependentina differentway[42-45] Forif(v1,…,vm)aresupersemivectorsfromthesuper semivectorspaceVoverR{0} L + ∪ .Wedeclarevi’sarelinearly dependentifvi= j aj Lv; ajR{0} LL + ∪ ∈ ,1 ≤ j ≤ m,j ≠ i,other wiselinearlyindependent. FortakeV= () {} 12345i aaaaaaR{0} LLLLLLL;1i5 + ∪ ∈≤≤ to beasupersemivectorspaceofrefinedlabelsoverR{0} L + ∪ . Consider(Lm+10|00|Lm+1)and( n a L0|00| n a L)inV.
Clearly( n a L0|00| n a L)= n a L(Lm+10|00|Lm+1)so,the giventwosupersemivectorsinVarelinearlydependentover R{0} L + ∪ .Nowconsider(Lm+10|00|0),(00|1 a L0|0)and(0 0|00| 1b L)inVweseethissetofsupersemivectorsinVare linearlyindependentinVoverR{0} L + ∪ .
Thuswiththissimpleconceptoflinearlyindependentsuper semivectorsinVwecandefineabasisofVasasetoflinearly independentsupersemivectorsinVwhichcangeneratethe supersemivectorspaceVoverR{0} L + ∪ . Consider V= () {} 123456i aaaaaaaR{0} LLLLLLLL;1i6 + ∪ ∈≤≤ tobeasupersemivectorspaceofrefinedlabelsoverR{0} L + ∪ . ConsiderthesetB={(Lm+1|0|00|00),(0|Lm+1|00|00),(0 |0|Lm+10|00),(0|0|0Lm+1|00),(0|0|00|Lm+10),(0|0| 00|0Lm+1)} ⊆ VisasetoflinearlyindependentelementsofV whichgenerateV.InfactBisabasisofthesupersemivector spaceVoverR{0} L + ∪ .Wecangivemoreexamples. Wewouldgivethedefinitionofsupercolumnvectorspace ofrefinedlabelsover R{0} L + ∪ . DEFINITION 5.2: Let
be an additive semigroup of super column vectors of refined labels. Clearly V is a super column semivector space of refined labels over LR.
173
V = + ∪ ∈≤≤ 1 2 3 i n1 n
a a R{0} a a L L L LL;1in ... L L
a a
Wewillillustratethissituationbysomeexamples.
Example 5.16: Let V=
L L L LL;1i6 L L L
1 2 3 i 4 5 6
a a a aR{0} a a a
+ ∪ ∈≤≤
beasupercolumnsemivectorspaceofrefinedlabelsover R{0} L + ∪ .
Example 5.17: Let V=
L L L L L L LLL;1i13 L L L L L L
1 2 3 4 5 6 7i 8 9 10 11 12 13
a a a a a a aaR{0} a a a a a a
+ ∪ ∈≤≤
beasupercolumnsemivectorspaceofrefinedlabelsover R{0} L + ∪ .
174
Example 5.18: Let
P=
L L L L L LLL;1i10 L L L L
1 2 3 4 5 i 6 7 8 9 10
a a a a a aQ{0} a a a a a
+ ∪ ∈≤≤
betheclassofsupercolumnsemivectorspaceofrefinedlabels over Q{0} L + ∪ .ItistobenotedPisnotasupercolumn semivectorspaceofrefinedlabelsoverR{0} L + ∪ .Asincaseof generalsemivectorspacesthedefinitiondependsonthe semifieldoverwhichthesemivectorspaceisdefined. Wewillillustratethissituationalsobysomeexamples.
Example 5.19: Let V=
a a a a a aR{0} a a a a a
1 2 3 4 5 i 6 7 8 9 10
175
L L L L L LL;1i10 L L L L L + ∪ ∈≤≤
beasupercolumnsemivectorspaceofrefinedlabelsoverthe semifield Q{0} L + ∪ .
Example 5.20: Let P=
L L L L
1 2 3 4 5i 6 7 8 9
a a a a aaQ{0} a a a a
+ ∪ ∈≤≤
LLL;1i9 L L L L
isasupercolumnsemivectorspaceofrefinedlabelsover Q{0} L + ∪ .
Nowwewillgiveexamplesofbasisanddimensionofthese spaces.
Example 5.21: Let V=
L L L L LL;1i8 L L L L
1 2 3 4 i 5 6 7 8
a a a a aQ{0} a a a a
+ ∪ ∈≤≤
beasupercolumnvectorspaceofrefinedlabelsoverQ{0} L + ∪ .
176
177 Consider m1 m1 m1 m1 m1 m1 L00000 0L0000 00L000 000L00 M,,0000L0 00000L 000000 000000 + + + + + + = m1 m1 00 00 00 00 00 00 L0 0L + + ⊆ VisabasisofVover Q{0} L + ∪ . Example 5.22: Let V= 1 2 3 4 5i 6 7 8 9 a a a a aaR{0} a a a a L L L L LLL;1i9 L L L L + ∪ ∈≤≤ beasupercolumnsemivectorspaceofrefinedlabelsover R{0} L + ∪ .ClearlyVisofdimensionnineoverR{0} L + ∪ . ConsiderVasasupercolumnsemivectorspaceofrefined labelsover Q{0} L + ∪ ,thenweseeVisofinfinitedimensionover Q{0} L + ∪ .Thisclearlyshowsthataswechangethesemifield overwhichthesupercolumnsemivectorspaceisdefinedis
changedthenweseethedimensionofthespacevariesasthe semifieldoverwhichVisdefinedvaries.
Example 5.23: Let V=
L L L LLL;1i6 L L
1 2 3 i 4 5 6
a a a aQ{0} a a a
+ ∪ ∈≤≤
beasupersemivectorspaceofrefinedlabelsoverQ{0} L + ∪ of dimensionsixover Q{0} L + ∪ .ClearlyVisnotdefinedoverthe refinedlabelfield R{0} L + ∪ .
Example 5.24: Let M=
L L LLL;1i5 L L
1 2 3i 4 5
a a aaR{0} a a
+ ∪ ∈≤≤
beasupercolumnsemivectorspaceofrefinedlabelsoverthe field R{0} L + ∪ . ClearlyMisofdimensionfiveoverR{0} L + ∪ ;butMisof dimensioninfiniteover Q{0} L + ∪ . Nowweproceedontogiveexamplesoflinear transformationsonsupercolumnsemivectorspaceofrefined labels.
178
179
L L L
L L L
L L L L L L LLL;1i13 L L L L L L + ∪ ∈≤≤ betwosupercolumnsemivectorspacesdefinedoverthesame semifield Q{0} L + ∪ . Defineamap η :V → Wgivenby
Example 5.25: Let V= 1 2 3 4i 5 6 7 a a a aaR{0} a a a
LLL;1i7
+ ∪ ∈≤≤ and W= 1 2 3 4 5 6 7i 8 9 10 11 12 13 a a a a a a aaR{0} a a a a a a
180 η 1 2 3 4 5 6 7 a a a a a a a L L L L L L L = 1 2 2 3 3 3 4 5 6 7 7 7 1 a a a a a a a a a a a a a L L L L L L L L L L L L L . Itiseasilyverified η isalineartransformationofsuper
Example 5.26: Let {} 12345678i aaaaaaaaaQ{0} VLLLLLLLLLL;1i7 + ∪ =∈≤≤ andW= 1 2 3 4 5i 6 7 8 9 a a a a aaQ{0} a a a a L L L L LLL;1i9 L L L L + ∪ ∈≤≤
semivectorspacesorsupersemivectorspacelinear transformationofVtoW.
181
L L L L L L L L 0
L L L LL;1i10 L L L L L
∈≤≤
betwosupersemivectorspacesofrefinedlabelsoverQ{0} L + ∪ . Let η:V → Wbedefinedby η ( () 12345678 aaaaaaaa LLLLLLLL)= 1 2 3 4 5 6 7 8 a a a a a a a a
. Itiseasilyverified η isasupersemivectorspacelinear transformationofVintoW. Example 5.27: Let P= 1 2 3 4 5 i 6 7 8 9 10 a a a a a aQ{0} a a a a a L L
+ ∪
beasupersemivectorspaceofrefinedlabelsoverQ{0} L + ∪ . ConsiderT:V → Vdefinedby
ontodefinesupermatrixsemivectorspaces.
182 T( 1 2 3 4 5 6 7 8 9 10 a a a a a a a a a a L L L L L L L L L L )= 1 2 4 5 3 6 a a a a a a L L 0 L 0 L L 0 L 0 ;
5.4: Let V = ++ ∪∪ ∈ 11121m 21222m i nnn 12m aaa aaa a R{0}Q{0} aaa LL...L LL...L LL(or L); LL...L 1 ≤ i ≤ n and 1 ≤ j ≤ m} be the additive semigroup of super matrices of refined labels. V is a super matrix semivector space of refined labels over {0} R L + ∪ . Example 5.28: Let M= 123 456 789 i 101112 131415 161718 aaa aaa aaa aR{0} aaa aaa aaa LLL LLL LLL LLLLL;1i18 LLL LLL + ∪ ∈≤≤
Tisalinearoperatorofsupersemivectorspaces.Nowhaving seenexamplesoflineartransformationsandlinearoperatorson supercolumnsemivectorspacesofrefinedlabels.Weproceed
DEFINITION
183 beasupermatrixsemivectorspaceofrefinedlabelsover R{0} L + ∪ .Misalsoknownasthesupercolumnvector semivectorspaceofrefinedlabelsoverthefieldofrefined labels R{0} L + ∪ . Example 5.29: Let V= 123 456i 789 aaa aaaaR{0} aaa LLL LLLLL;1i9 LLL + ∪ ∈≤≤ bethesupersquarematrixsemivectorspaceofrefinedlabels over R{0} L + ∪ . Example 5.30: Let 123456789 101112131415161718 192021222324252627 282930313233343536 aaaaaaaaa aaaaaaaaa aaaaaaaaa aaaaaaaaa LLLLLLLLL LLLLLLLLL W LLLLLLLLL LLLLLLLLL = } i aR{0} LL;1i36 + ∪ ∈≤≤ beasuperrowvectorsemivectorspace ofrefinedlabelsover R{0} L + ∪ orsupermatrixsemivectorspace ofrefinedlabelsover R{0} L + ∪ . Example 5.31: Let 2345678 910111213141516 1718192021222324 2526272829303132 3334353637383940 4142434445464748 aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa LLLLLLLL LLLLLLLL LLLLLLLL V LLLLLLLL LLLLLLLL LLLLLLLL =
} i aR{0} LL;1i48 + ∪ ∈≤≤ beasupermatrixvectorspaceofrefined labelsoverthesemifieldofrefinedlabels R{0} L + ∪ . Weasincaseofothersupersemivectorspacesdefinethe notionofsupersemivectorsubspaces,basis,dimensionand lineartransformations.Howeverthistaskisdirectandhenceis leftasanexercisetothereaderweonlygiveexamplesofthem.
Example 5.32: Let V=
LLL LLL LLL LL;1i18 LLL LLL LLL
123 456 789 i 101112 131415 161718
aaa aaa aaa aR{0} aaa aaa aaa
+ ∪ ∈≤≤
beasupermatrixsemivectorspaceorsupercolumnvector semivectorspaceoverR{0} L + ∪ .Clearly P=
000 LLL 000 LL;1i9 LLL 000 LLL
123 i 456 789
aaa aR{0} aaa aaa
+ ∪ ∈≤≤⊆ V
isasupercolumnvector,semivectorsubspaceofrefinedlabels ofVover R{0} L + ∪ .Wehaveseveralbutonlyfinitenumberof supermatrixsemivectorsubspacesofVoverR{0} L + ∪ .IfVbe definedovertherefinedsemifieldoflabels Q{0} L + ∪ thenVhas infinitenumberofsupermatrixsemivectorsubspaces.Thus eventhenumberofsupermatrixsemivectorsubspacesdepends onthesemifieldoverwhichitisdefined.Thisisevidentfrom example5.32.
184
Example 5.33: Let V=
LLLLLLL LLLLLLL LLLLLLLLL; LLLLLLL1i42 LLLLLLL LLLLLLL
1234567 891011121314 15161718192021i 22232425262728 29303132333435 36373839404142
aaaaaaa aaaaaaa aaaaaaaaQ{0} aaaaaaa aaaaaaa aaaaaaa
+ ∪ ∈ ≤≤ beasupermatrixofsemivectorspaceofrefinedlabelsover Q{0} L + ∪ .Visafinitedimensionalsupermatrixsemivector subspaceoverQ{0} L + ∪ havinginfinitenumberofsemivector subspacesofrefinedlabelsoverQ{0} L + ∪ .
Nowweproceedontodefinenewclassofsupermatrix semivectorspacesovertheintegersemifieldZ+ ∪ {0};known astheintegersupermatrixsemivectorspaceofrefinedlabels. Wewillillustratethembeforeweproceedontoderive propertiesrelatedwiththem.
Example 5.34: Let V= () {} 1234567i aaaaaaaaQ{0} LLLLLLLLL;1i7 + ∪ ∈≤≤ beaintegersuperrowsemivectorspaceofrefinedlabelsover Z+ ∪ {0},thesemifieldofintegers.
185
Example 5.35: Let V= 123 456 789i 101112 131415 aaa aaa aaaaR{0} aaa aaa LLL LLL LLLLL;1i15 LLL LLL + ∪ ∈≤≤
186
V= 1 2 3 4 i 5 6 7 8 a a a a aR{0} a a a a L L L L LL L L L L + ∪ ∈
V= 12345678 910111213141516 1718192021222324 2526272829303132 3334353637383940 aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa LLLLLLLL LLLLLLLL LLLLLLLL LLLLLLLL LLLLLLLL } i aQ{0} LL;1i40 + ∪ ∈≤≤ beaintegersupermatrixsemivector spaceofrefinedlabelsoverthesemifieldZ+ ∪ {0}. Wecandefinesubstructures,lineartransformationabasis; thistaskcanbedoneasamatterofroutinebytheinterested reader.
beanintegersupermatrixsemivectorspaceofrefinedlabels overZ+ ∪ {0},thesemifieldofintegers. Example 5.36: Let
beanintegersuperrowsemivectorspaceoverthesemifieldZ+ ∪ {0}. Example 5.37: Let
Butweillustrateallthesesituationsbysomeexamples.
Example 5.38: Let M=
L L L L L LL;1i10 L L L L L
1 2 3 4 5 i 6 7 8 9 10
a a a a a aQ{0} a a a a a
+ ∪ ∈≤≤
beanintegersupercolumnsemivectorspaceofrefinedlabels overthesemifieldZ+ ∪ {0}. Take K=
1 2 3 i 4 5
L 0 L 0 L LL;1i5 0 L 0 L 0
a a a aQ{0} a a
+ ∪ ∈≤≤⊆ M,
Kisaintegersupercolumnsemivectorsubspacesofrefined labelsoverthesemifieldZ+ ∪ {0}.
Infactwecanhaveinfinitenumberofintegersupercolumn semivectorsubspacesofVofrefinedlabelsoverZ+ ∪ {0}. Fortake
187
isagainanintegersupercolumnsemivectorsubspaceofMof refinedlabelsoverZ+ ∪ {0}.Infact6canbereplacedbyany positiveintegerinZ+andKwillcontinuetobeasupercolumn semivectorsubspaceofMofrefinedlabelsoverZ+ ∪ {0}.
Example 5.39: Let P= () {} 123456i aaaaaaaR{0} LLLLLLLL;1i6 + ∪ ∈≤≤ beanintegersuperrowsemivectorspaceofrefinedlabelover thesemifieldZ+ ∪ {0}. K= () {} 1234i aaaaaR{0} LLL00LLL;1i4 + ∪ ∈≤≤⊆ Pis anintegersuperrowsemivectorsubspaceofPofrefinedlabels overtheintegersemifieldZ+ ∪ {0}.
188
P= 1 2 3 4 5 i 6 7 8 9 10 a a a a a aT a a a a a L L L L L LLLwhere6Z{0};1i10 L L L L + ∈∪≤≤⊆ M
Example 5.40: Let S= 123 456 789i 101112 131415 aaa aaa aaaaQ{0} aaa aaa LLL LLL LLLLL;1i15 LLL LLL + ∪ ∈≤≤
189 beanintegersupermatrixsemivectorspaceofrefinedlabels
+
Consider H= 1 2 34i 56 78 a a aaaR{0} aa aa 00L 00L LL0LL;1i8 LL0 LL0 + ∪ ∈≤≤⊆ S; HisanintegersupermatrixsemivectorsubspaceofSoverthe
+ ∪
ItispertinenttomentionherethatShasinfinitelymany
Example 5.41:Let V= () {} 123456i aaaaaaaR{0} LLLLLLLL;1i6 + ∪ ∈≤≤ beanintegersuperrowvectorsemivectorspaceofrefined
+
Take W1= () {} 12i aaaR{0} LL0000LL;1i2 + ∪ ∈≤≤⊆ V, W2= () {} 1i aaR{0} 00L000LL + ∪ ∈⊆ V, W3= () {} 12i aaaR{0} 000LL0LL;1i2 + ∪ ∈≤≤⊆ V and W4= () {} 1i aaR{0} 00000LLL + ∪ ∈⊆ V beintegersuperrowvectorsemisubspacesofV. ClearlyV= 4 i i1 W = ;withWi ∩ Wj=(0|000|00);i ≠ j,1 ≤ i,j ≤ 4.Visthedirectunionofintegersuperrowsemivector subspacesofVoverthesemifieldZ+ ∪ {0}.
overthesemifieldZ
∪ {0}.
integersemifieldZ
{0}.
integersupermatrixsemivectorsubspaceofrefinedlabelsover theintegersemifieldZ+ ∪ {0}.
labelsoverthesemifieldofintegersZ
∪ {0}.
Example 5.42: Let
V=
LLL LLL LLL LLL LL;1i24 LLL LLL LLL LLL
123 456 789 101112 i 131415 161718 192021 222324
aaa aaa aaa aaa aR{0} aaa aaa aaa aaa
+ ∪ ∈≤≤
beanintegersupercolumnvectorsemivectorspaceofrefined labelsoverthesemifieldZ+ ∪ {0}=S.Consider V1=
123 i
aaa aR{0}
LLL 000 000 000 LL;1i3 000 000 000 000
+ ∪ ∈≤≤⊆ V, V2=
123 456 i
aaa aaa aR{0}
000 LLL LLL 000 LL;1i6 000 000 000 000
+ ∪ ∈≤≤⊆ V,
190
191
V and
∪
V beintegersupercolumnvectorsemivectorsubspacesofVover thesemifieldS=Z+ ∪ {0}.ClearlyV= 4 i i1 V = with Vi ∩ Vj= 000 000 000 000 000 000 000 000 ifi ≠ j,1 ≤ i,j ≤ 4.
V3= 123 i 456 789 aaa aR{0} aaa aaa 000 000 000 LLL LL;1i9 LLL LLL 000 000 + ∪ ∈≤≤⊆
V4= i 123 456 aR{0} aaa aaa 000 000 000 000 LL;1i6 000 000 LLL LLL +
∈≤≤⊆
192
V= 123 456 789 i
131415 161718 aaa
aR{0} aaa aaa
LLL LLL LLL LL;1i18 LLL LLL LLL + ∪ ∈≤≤
+
P
12 3 4 i 6 5 aa a a aR{0} a a LL0 0L0 L00 LL;1i6 00L 000 0L0 + ∪ ∈≤≤⊆ V, P
12 34 i 6 5 aa aa aR{0} a a L0L 000 LL0 LL;1i6 00L 000 L00 + ∪ ∈≤≤⊆ V,
ThusVisadirectsumofintegersupercolumnvector semivectorsubspacesofVoverS. Example 5.43: Let
101112
aaa aaa
aaa
beanintegersupermatrixsemivectorspaceofrefinedlabels overthesemifieldS=Z
∪ {0}. Consider
1=
2=
193
P3= 1 423 i 56 a aaa aR{0} aa L00 LLL 000 LL;1i6 000 000 LL0 + ∪ ∈≤≤⊆ V, P4= 1 24 i 5 7 6 a aa aR{0} a a a L00 000 L0L LL;1i7 L00 0L0 L00 + ∪ ∈≤≤⊆ V, P5= 1 i 2 3 a aR{0} a a L00 000 000 LL;1i3 000 L00 L00 + ∪ ∈≤≤⊆ V andP6= 12 i 3 4 5 aa aR{0} a a a LL0 000 000 LL;1i5 0L0 00L 00L + ∪ ∈≤≤⊆ V areintegersupermatrixsemivectorsubspacesofrefinedlabels ofVoverthesemifieldZ+ ∪ {0}.
194 ClearlyV= 6 i i1 P = andVi ∩ Vj ≠ 000 000 000 000 000 000 ifi ≠ j,1≤i,j ≤ 6. ThusVisapseudodirectsumofintegersupermatrix semivectorsubspacesofVoverZ+ ∪ {0}ofrefinedlabels. Example 5.44: Let B= 16111213 27141516 38171819i 49202122 510232425 aaaaa aaaaa aaaaaaR{0} aaaaa aaaaa LLLLL LLLLL LLLLLLL;1i25 LLLLL LLLLL + ∪ ∈≤≤ beaintegersuperrowvectorsemivectorspaceofrefinedlabels overthesemifieldZ+ ∪ {0}. Consider H1= 1 2 36i 4 5 a a aaaR{0} a a L0000 L0000 L000LLL;1i6 L0000 L0000 + ∪ ∈≤≤⊆ B, H2= 1234 5 6i 7 aaaa a aaR{0} a LLLL0 0000L 0000LLL;1i7 0000L 00000 + ∪ ∈≤≤⊆B,
H3=
0L000 LLLLL 00000LL;1i6 00000 00000
2 14356 i
a aaaaa aR{0}
+ ∪ ∈≤≤⊆ B, H4=
L000L L000L 0LLL0LL;1i10 00000 0LLL0
19 210 345i 678
aa aa aaaaR{0} aaa
+ ∪ ∈≤≤⊆ B, and H5=
0L000 L0L00 000L0LL;1i9 0LLL0 000LL
2 13 4i 567 89
a aa aaR{0} aaa aa
+ ∪ ∈≤≤⊆ B, beanintegersuperrowvectorsemivectorsubspacesofBof refinedlabelsovertheintegersemifieldS=Z+ ∪ {0}. B= 5 i i1 H =
and Hi ∩ Hj ≠
00000 00000 00000 00000 00000
ifi ≠ j,1 ≤ i,j ≤ 5.
ThusHisonlyapseudodirectsumofintegersuperrow vectorsemivectorsubspacesofBofrefinedlabelsoverZ+ ∪ {0}.
195
Nowhavingseenexamplesofpseudodirectsumofinteger supermatrixsemivectorsubspacesanddirectsumofinteger supermatrixsemivectorsubspaceswenowproceedontogive examplesofintegerlineartransformationandintegerlinear operatorofintegersupermatrixsemivectorspacesdefinedover theintegersemifieldZ+ ∪ {0}.
Example 5.45: Let V=
LLL LLL LLL LLLLL;1i21 LLL LLL LLL
123 456 789 101112i 131415 161718 192021
aaa aaa aaa aaaaQ{0} aaa aaa aaa
+ ∪ ∈≤≤
LLLLLLLL LL; LLLLLLLL 1i24 LLLLLLLL
beanintegersupermatrixsemivectorspaceofrefinedlabels overtheintegersemifieldZ+ ∪ {0}. W= 1471013161922 i 2581114172023 3691215182124
aaaaaaaa aR{0} aaaaaaaa aaaaaaaa
+ ∪ ∈ ≤≤
beanintegersupermatrixsemivectorspaceofrefinedlabels overtheintegersemifieldZ+ ∪ {0}.DefineTaintegerlinear transformationfromVintoWasfollows. T
LLL LLL LLL LLL LLL LLL LLL
aaa aaa aaa aaa aaa aaa aaa
123 456 789 101112 131415 161718 192021
196
LLLLLLL0 LLLLLLL0 LLLLLLL0 .
= 14710131619 25811141720 36912151821
aaaaaaa aaaaaaa aaaaaaa
Example 5.46: Let V=
LLL LLL LLL LLL LLLLL;1i27 LLL LLL LLL LLL
123 456 789 101112 131415i 161718 192021 222324 252627
aaa aaa aaa aaa aaaaQ{0} aaa aaa aaa aaa
+ ∪ ∈≤≤ beanintegersupersemivectorspaceofrefinedlabelsoverthe semifieldS=Z+ ∪ {0}. DefineT:V → Vby T
LLL LLL LLL LLL LLL LLL LLL LLL LLL
aaa aaa aaa aaa aaa aaa aaa aaa aaa
123 456 789 101112 131415 161718 192021 222324 252627
=
000 LLL 000 LLL 000 LLL 000 LLL 000
aaa aaa aaa aaa
123 456 789 101112
.
197
ClearlyTisaintegerlinearoperatoronVwherekernelTis anontrivialsemivectorsubspaceofV.
Nowwegiveanexampleofaprojectiontheconceptis directandcandefineitasincaseofsemivectorspaces.
Example 5.47: Let V=
LLL LLL LLL LL;1i18 LLL LLL LLL
123 456 789 i 101112 131415 161718
aaa aaa aaa aQ{0} aaa aaa aaa
+ ∪ ∈≤≤
beanintegersupermatrixsemivectorspaceofrefinedlabels overtheintegersemifieldS=Z+ ∪ {0}. Consider W=
0L0 LLL 00L LL;1i9 LL0 000 LL0
1 234 5 i 67 89
a aaa a aQ{0} aa aa
+ ∪ ∈≤≤⊆ V
beanintegersupermatrixsemivectorspaceofrefinedlabels overthesemifieldZ+ ∪ {0}ofV.
DefineTanintegerlinearoperatorfromVintoVgivenby
T
LLL LLL LLL LLL LLL LLL
aaa aaa aaa aaa aaa aaa
123 456 789 101112 131415 161718
=
0L0 LLL 00L LL0 000 LL0
a aaa a aa aa
1 234 5 67 89
.
198
ClearlyTisalinearoperatorwithnontrivalkernel. FurtherT(W) ⊆ WthatisWisinvariantundertheinteger linearoperatorTonV.Consider
T 1(
LLL LLL LLL LLL LLL LLL
aaa aaa aaa aaa aaa aaa
123 456 789 101112 131415 161718
)=
LLL 000 LLL 000 LLL 000
aaa aaa aaa
123 456 789
.
WeseeT 1isaintegerlinearoperatorwithnontrivialkernel butclearlyT 1isnotanintegerlinearoperatorwhichkeepsW invariant;thatisT 1(W) ⊄ W. ThusallintegerlinearoperatorsingeneralneednotkeepW invariant.
Nowweproceedontodefinevarioustypesofsupermatrix semivectorspacesofrefinedlabels.
DEFINITION 5.4: Let V = {set of super matrices whose entries are labels from {0} R L + ∪ or {0} Q L + ∪ } be a set of super matrices refined labels. If V is a such that for a set S subset of positive reals (R+ ∪ {0} or Z+ ∪ {0} or Q+ ∪ {0}) sv and vs are in V for every v ∈ V and s ∈ S; then we define V to be a set super matrix semivector space of refined labels over the set S or super matrix semivector space of refined labels over the set S, of subset of reals.
Wewillfirstillustratethissituationbysomeexamples.
199
Example 5.48: Let V= () 1 212 12345678 334 4 a aaa aaaaaaaa aaa a L LLL ,,LLLLLLLL LLL L
semivectorspaceofrefinedlabelsoverthesetS.
∈ 3Z+ ∪ 25Z+ ∪ {0}} ⊆ R{0} L + ∪ ofrefinedlabels. ClearlyVisofinfiniteorderwecandefinetwosubstructure forV,asupermatrixsetsemivectorspaceofrefinedlabelsover thesetS,S ⊆ R+ ∪ {0}.
DEFINITION 5.5: Let V be a super matrix semivector space of refined labels with entries from {0} R L + ∪ over the set S (S ⊆ R+ ∪ {0}). Suppose W ⊆ V, W a proper subset of V and if W itself is a set super matrix semivector space over the set S then we define W to be a set super matrix semivector subspace of V of refined labels over the set S. Wewillfirstillustratethissituationbysomeexamples.
200
} i aQ{0} LL;1i8 + ∪ ∈≤≤ beasetofsupermatricesofrefined labels. LetS={a|a ∈ 3Z+ ∪ 2Z+ ∪ {0}.Visasetsupermatrix
Example 5.49: Let V= 123147101316 456258111417 789369121518 aaaaaaaaa aaaaaaaaa aaaaaaaaa LLLLLLLLL LLL,LLLLLL, LLLLLLLLL () 1 2 3 i 412345678 5 6 7 a a a aQ{0} aaaaaaaaa a a a L L L LLL; LLLLLLLL L1i18 L L + ∪ ∈ ≤≤ beasupermatrixsetsemivectorspaceoverthesetS={a|a
201 Example 5.50:Let 12123 34456 123456 56789 789101112 78101112 910131415 aaaaa aaaaa aaaaaa aaaaa aaaaaa aaaaa aaaaa LLLLL LLLLL LLLLLL LLLLLV,, LLLLLL LLLLL LLLLL = } i aQ{0} LL;1i15 + ∪ ∈≤≤ beasetsupermatrixsemivectorspace ofrefinedlabelsoverthesetS={a ∈ 3Z+ ∪ 5Z+ ∪ {0}} ⊆ Q+ ∪ {0}.Consider W= 1216 3427 38i 49 56510 aaaa aaaa aaaQ{0} aa aaaa LLL0L LLL0L 00L0L,LL;1i10 00L0L LLL0L + ∪ ∈≤≤ ⊆ V,Wisasetsupermatrixsemivectorsubspaceofrefined labelsoverthesetS ⊆ Q+ ∪ {0}. Example 5.51: Let 1 212345613 378910111214 415161718192021 522232425262728 629303132333435 7 a aaaaaaaa a aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa a L LLLLLLLL L LLLLLLLL LLLLLLLL V,, LLLLLLLL LLLLLLLL L = 1237 4568 9101112 13141516 aaa aaaa aaaa aaaa LLL LLLL LLLL LLLL } i aQ{0} LL;1i35 + ∪ ∈≤≤ beasetsupermatrixsemivectorspace ofrefinedlabelsoverthesetS=3Z+ ∪ 5Z+ ∪ 8Z+ ∪ {0}} ⊆ Q+ ∪ {0}.
subspaceofVoverthesetSofrefinedlabels.
Nowweproceedontodefinethenotionofsubsetsuper semivectorsubspaceofrefinedlabels.
DEFINITION 5.6: Let V be a set super matrix semivector space of refined labels over the set S, (S ⊆ R+ ∪ {0}). Let W ⊆ V; and P ⊆ S (W and P are proper subsets of V and S respectively). If W is a super matrix semivector space of refined labels over the set P then we define W to be a subset super matrix semivector subspace of refined labels over the subset P of the set S. Wewillillustratethissituationbysomeexamples.
202
1
12 21891012 86
a
aaaaaa
aaa aaaa aaaaa L 000L000L L00L LL00LLLL 0L0L 00LL0000P,, 00LL L0L000L0 LLLL LL0LL00L 0 = } i aQ{0} LL;1i15 + ∪ ∈≤≤⊆ V;Pisasetsupermatrixsemivector
Consider
511
26 97 3313 34516 4471514
aa aa
aa aa aa
Let P= 123 456 789147101316 101112258111417 131415369121518 161718 192021 aaa aaa aaaaaaaaa aaaaaaaaa aaaaaaaaa aaa aaa LLL LLL LLLLLLLLL LLL,LLLLLL, LLLLLLLLL LLL LLL
Example 5.52:
KisasubsetsupermatrixsemivectorsubspaceofPoverthe subsetTofS.
203
123 456 i 789 101112 aaa aaa aQ{0} aaa aaa LLL LLL LL;1i21 LLL LLL + ∪ ∈≤≤
beasetsupermatrixsemivectorspaceofrefinedlabelsoverthe set S={a ∈ 3Z+ ∪ 8Z+ ∪ 7Z+ ∪ 13Z+ ∪ {0}} ⊆ Q+ ∪ {0}. Consider K= 123 456137 458 789269 111213 aaa aaaaaa aaa aaaaaa aaa LLL 000 LLLL0L0L0 000,0L0L0L, LLLL0L0L0 000 LLL 123 456 i 789 101112 aaa aaa aQ{0} aaa aaa LLL LLL LL;1i12 LLL LLL + ∪ ∈≤≤⊆ P and T=a ∈ 3Z+ ∪ 7Z+ ∪ {0}} ⊆ S ⊆ Q+ ∪ {0}.
204 Example 5.53: Let () 1 2 3 4 5123456789 6 7 8 9 a a a a aaaaaaaaaa a a a a L L L L L V,LLLLLLLLL, L L L L = 12 123434 567856 i 910111278 13141516910 1112 aa aaaaaa aaaaaa aQ{0} aaaaaa aaaaaa aa LL LLLLLL LLLLLL LLLL,LL;1i16 LL LLLLLL LL + ∪ ∈≤≤ beasetsupermatrixsemivectorspaceofrefinedlabelsoverthe setS=a ∈ 5Z+ ∪ 3Z+ ∪ 7Z+ ∪ 8Z+ ∪ 13Z+ ∪ 11Z+ ∪ {0}}. Consider W= () 1 2 12345 3 4 5 a a aaaaa a a a 0 L 0 L 0 L00LLL0L0,, 0 L L L
205 1 212 3434 i 5 656 7 a aaa aaaa aQ{0} a aaa a L0 0LL0L0 LL0L0L ,LL;1i7L00000 0LL0L0 L0 + ∪ ∈≤≤⊆ V andK=a ∈ 3Z+ ∪ 8Z+ ∪ 13Z+ ∪ {0}} ⊆ S ⊆ Q+ ∪ {0}.
Example 5.54: Let 1 2 31234 45678 147101316 59101112 2 613141516 717181920 821222324 9 10 a a aaaaa aaaaa aaaaaa aaaaa a aaaaa aaaaa aaaaa a a L L LLLLL LLLLL LLLLLL LLLLL V,,L LLLLL LLLLL LLLLL L L = 58111417 369121518 aaaaa aaaaaa LLLLL LLLLLL } i aQ{0} LL;1i24 + ∪ ∈≤≤ beasetsupermatrixofsemivector spaceofrefinedlabelsoverthesetS={a ∈ 3Z+ ∪ 25Z+ ∪ 11Z+ ∪ 32Z+ ∪ {0}} ⊆ Q+ ∪ {0}.
ClearlyWisasubsetsupermatrixsemivectorsubspaceofVof refinedlabelsoverthesubsetKofS. Asincaseofusualvectorspaceswecandefinedirectunion andpseudodirectunionofsetsupermatrixsemivectorspace. Thedefinitioniseasyanddirectandhenceisleftforthereader asanexercise. Wenowillustratethesetwosituationsbysomeexamples.
206 Consider W1= 1 2 3 4 5 i 6 7 8 9 10 a a a a a aQ{0} a a a a a L L L L L LLL;1i10 L L L L + ∪ ∈≤≤⊆ V, W2= 147101316 258111417i 369121518 aaaaaa aaaaaaaQ{0} aaaaaa LLLLLL LLLLLLLL;1i18 LLLLLL + ∪ ∈≤≤ ⊆ Vand W3= 1234 5678 9101112 i 13141516 17181920 21222324 aaaa aaaa aaaa aQ{0} aaaa aaaa aaaa LLLL LLLL LLLL LLLLLL;1i24 LLLL LLLL + ∪ ∈≤≤⊆ V besetsupermatrixsemivectorsubspacesofrefinedlabelsover thesetS. NowV= 3 i i1 W = andWi ∩ Wj= φ ifi ≠ j,1 ≤ i,j ≤ 3.Thus Visthedirectunionofsetsupermatrixsemivectorsubspaces.
207 Example 5.55: Let 1 2 3 4 5 6 7 135791113 8 2468101214 9 10 11 12 13 14 15 a a a a a a a aaaaaaa a aaaaaaa a a a a a a a L L L L L L L LLLLLLL L V, LLLLLLL L L L L L L L = 123 1234456 5678789i 9101112101112 13141516131415 161718 aaa aaaaaaa aaaaaaaaQ{0} aaaaaaa aaaaaaa aaa LLL LLLLLLL LLLLLLLLL; LLLL,LLL1i18 LLLLLLL LLL + ∪ ∈ ≤≤ beasetsupermatrixsemivectorspaceofrefinedlabelsoverthe setS={3Z+ ∪ 7Z+ ∪ 13Z+ ∪ 11Z+ ∪ 19Z+ ∪ 8Z+ ∪ {0}} ⊆ Q+ ∪ {0}. Consider
208 1 2 3 4 5 6 7 13i 8 24 9 10 11 12 13 14 15 a a a a a a a aaaQ{0} 1a aa a a a a a a a L L L L L L L L00000LLL; L M, L00000L1i15 L L L L L L L + ∪ ∈ = ≤≤ ⊆ V, M2= 14578910i 23611121314 aaaaaaaaQ{0} aaaaaaa LLLLLLLLL; LLLLLLL1i14 + ∪ ∈ ≤≤ ⊆ V, M3= 1234 5678135 9101112246 13141516 aaaa aaaaaaa aaaaaaa aaaa LLLL LLLLLL0000L , LLLLLL0000L LLLL } i aQ{0} LL;1i16 + ∪ ∈≤≤⊆ V and
butMi ∩ Mj ≠φ ifi ≠ j,1 ≤ i,j ≤ 4.
besetsupermatrixsemivectorsubspacesofrefinedlabelsofV overthesetS ⊆ Q+ ∪ {0}. ClearlyV= 4 i i1 M =
ThusVisonlyapseudodirectunionofsetsupermatrices semivectorsubspaceofrefinedlabelsoverS. Wecandefinesetlineartransformationofsetsupermatrix semivectorspacesonlyifboththesetsupermatrixsemivector spaceofrefinedlabelsaredefinedoverthesamesetS ⊆ R+ ∪ {0}.
209
M4= 123 456 789 i 101112 131415 161718 aaa aaa aaa aQ{0} aaa aaa aaa LLL LLL LLL LLLLL;1i18 LLL LLL + ∪ ∈≤≤ and 15 j 2346 aa aQ{0} aaaa L00000L LL00L0LLL;1j16 + ∪ ∈≤≤⊆ V
Let () 1815 2916 31017 4111812345678 51219 61320 71421 aaa aaa aaa aaaaaaaaaaa aaa aaa aaa LLL LLL LLL LLL V,LLLLLLLL, LLL LLL LLL =
Wewilljustillustratethissituationbysomeexamples. Example 5.56:
LLLLL LLLLL LLLLLLL;1i25 LLLLL LLLLL
12345 678910 1112131415i 1617181920 2122232425
aaaaa aaaaa aaaaaaQ{0} aaaaa aaaaa
+ ∪ ∈≤≤
beasetsupermatrixsemivectorspaceofrefinedlabelsoverS= a ∈ 5Z+ ∪ 3Z+ ∪ {0}} ⊆ Q+ ∪ {0}.Let 14710131619 1234 25811141720 5678 36912151821
aaaaaaa aaaa aaaaaaa aaaa aaaaaaa
LLL LLL LLL LLL LL;1i24 LLL LLL LLL LLL
aaa aaa aaa aaa aQ{0} aaa aaa aaa aaa
+ ∪ ∈≤≤
LLLLLLL LLLL W,LLLLLLL, LLLL LLLLLLL = 123 456 789 101112 i 131415 161718 192021 222324
beasetsupermatrixsemivectorspaceofrefinedlabelsoverS= {a ∈ 5Z+ ∪ 3Z+ ∪ {0}} ⊆ Q+ ∪ {0}.DefineT:V → Waset lineartransformationofrefinedlabelsetsupersemivector spacesasfollows. T(
LLL LLL LLL LLL LLL LLL LLL
aaa aaa aaa aaa aaa aaa aaa
123 456 789 101112 131415 161718 192021
)
210
211 =
LLLLLLL LLLLLLL , T () () 12345678 aaaaaaaa LLLLLLLL=
LLLL LLLL and T(
aaaaa aaaaa aaaaa aaaaa aaaaa LLLLL LLLLL LLLLL LLLLL LLLLL = 123
LLL LLL LLL LLL LLL LLL LLL LLL . Tisasetsuperlineartransformationofsetsuperlinear semivectorspacesofrefinedlabelsdefinedoverthesetS ⊆ Q+ ∪ {0}. Example 5.57: Let V= 1815 291612345 31017678910 411181112131415 512191617181920 6132021222324 71421 aaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaa aaa LLL LLLLLLLL LLLLLLLL LLLLLLLL , LLLLLLLL LLLLLLL LLL 25 a , L
14710131619 25811141720 36912151821 aaaaaaa aaaaaaa aaaaaaa LLLLLLL
1234 5678 aaaa aaaa
12345 678910 1112131415 1617181920 2122232425
456 789 101112 131415 161718 192021 222324 aaa aaa aaa aaa aaa aaa aaa aaa
212 1 2 3 4 514710131619 i 625811141720 736912151821 23 24 25 a a a a aaaaaaaa aQ{0} aaaaaaaa aaaaaaaa a a a L L L L LLLLLLLL LL; L,LLLLLLL 1i25 LLLLLLLL L L L + ∪ ∈ ≤≤ beasetsupermatrixsemivectorspaceofrefinedlabelsoverthe setS={a ∈ 3Z+ ∪ {0} ∪ 2Z+}. DefineamapT:V → Vgivenby T 1815 2916 31017 41118 51219 61320 71421 aaa aaa aaa aaa aaa aaa aaa LLL LLL LLL LLL LLL LLL LLL = 14710131619 25811141720 36912151821 aaaaaaa aaaaaaa aaaaaaa LLLLLLL LLLLLLL LLLLLLL
213
T
L
L L L L L
T 12345 678910 1112131415 1617181920 2122232425 aaaaa aaaaa aaaaa aaaaa aaaaa LLLLL LLLLL LLLLL LLLLL LLLLL = 1 2 3 4 5 6 7 23 24 25 a a a a a a a a a a L L L L L L L L L L ,
1 2 3 4 5 6 7 23 24 25 a a a a a a a a a a
L L L L
= 12345 678910 1112131415 1617181920 2122232425 aaaaa aaaaa aaaaa aaaaa aaaaa LLLLL LLLLL LLLLL LLLLL LLLLL and T 14710131619 25811141720 36912151821 aaaaaaa aaaaaaa aaaaaaa LLLLLLL LLLLLLL LLLLLLL
ClearlyTisasetsupermatrixsemivectorspacesetlinear operatorofrefinedlabels. Asincaseofusualvectorspaceswecantalkofasetsuper matrixsemivectorsubspaceofrefinedlabelswhichisinvariant underasetsuperlinearoperatorTofV. Example 5.58: Let
214
= 1815 2916 31017 41118 51219 61320 71421 aaa aaa aaa aaa aaa aaa aaa LLL LLL LLL LLL LLL LLL LLL .
V=
2 3
4
5
6
7 8 9
a a a L L L LLLL L LLLLLL L ,,, LLLLLL L LLLL L L L 14710131619 25811141720i 36912151821 aaaaaaa aaaaaaaaQ{0} aaaaaaa LLLLLLL LLLLLLLLL;1i21 LLLLLLL + ∪ ∈≤≤
+ ∪
+
+
1
1234
567813
910111224
13141516
a a a aaaa a aaaaaa a aaaaaa a aaaa
beasetsupermatrixsemivectorspaceofrefinedlabelsover thesetS=a ∈ 3Z
2Z
∪ 7Z
∪ {0}} ⊆ Q+ ∪ {0}.
215 Consider W= 1 2 3 1234 4 135678i 5 249101112 6 13141516 7 8 9 a a a aaaa a aaaaaaaQ{0} a aaaaaa a aaaa a a a L L L LLLL L LLLLLLLLL; ,, LLLLLLL1i16 LLLL L L L + ∪ ∈ ≤≤ ⊆ VisasetsupermatrixsemivectorsubspaceofVofrefined labelsoverthesetS.Defineasetsuperlineartransformation T:V → Vby T 1 2 3 4 5 6 7 8 9 a a a a a a a a a L L L L L L L L L = 1 2 3 4 5 6 7 8 9 a a a a a a a a a L L L L L L L L L T13 24 aa aa LL LL = 13 24 aa aa LL LL,
T
LLLL LLLL LLLL LLLL
aaaa aaaa aaaa aaaa
1234 5678 9101112 13141516
=
LLLL LLLL LLLL LLLL and T 14710131619 25811141720 36912151821
aaaa aaaa aaaa aaaa
1234 5678 9101112 13141516
aaaaaaa aaaaaaa aaaaaaa
LLLLLLL LLLLLLL LLLLLLL = 0000000 0000000 0000000 .
TisasetlinearsuperoperatoronV. FurtherT(W) ⊆ W.ThusWisinvariantundertheset linearsuperoperatorofV.
Nowweproceedontodefinethenotionofsemigroupsuper matrixsemivectorspaceofrefinedlabelsoverasemigroupS ⊆ R+ ∪ {0}.
DEFINITION 5.7: Let V be a set super matrix semivector space of refined labels over the set S ⊆ R+ ∪ {0}.
If S is a semigroup of refined labels contained in R+ ∪ {0} then we call the set super matrix semivector space of refined labels as semigroup super matrix semivector space of refined labels over the semigroup S ⊆ R+ ∪ {0}.
Wewillillustratethissituationbysomeexamples.
216
217 Example 5.59: Let V= 12 34 56 12345 78 678910 910 1112131415 1112 1617181920 1314 1516 1718 aa aa aa aaaaa aa aaaaa aa aaaaa aa aaaaa aa aa aa LL LL LL LLLLL LL LLLLL LL ,, LLLLL LL LLLLL LL LL LL ()} 1234567i aaaaaaaaR{0} LLLLLLLLL;1i20 + ∪ ∈≤≤ beasemigroupsupermatrixsemivectorspaceofrefinedlabels overthesemigroupS=3Z+ ∪ {0}. Example 5.60: Let M= () 12 34 12345678 56 78 aa aa aaaaaaaa aa aa LL LL ,LLLLLLLL, LL LL 123456 789101112 131415161718i 192021222324 252627282930 aaaaaa aaaaaa aaaaaaaQ{0} aaaaaa aaaaaa LLLLLL LLLLLL LLLLLLLL;1i30 LLLLLL LLLLLL + ∪ ∈≤≤
isasemigroupsupermatrixsemivectorspaceofrefinedlabels overthesemigroupZ+ ∪ {0}underaddition.
Itispertinenttomentionherethatthesemigrioupmustbeonly asubsetofR+ ∪ {0}butitcanbeasemigroupunderadditionor undermultiplication.
beasemigroupsupermatrixsemivectorspaceoverthe semigroupQ{0} L + ∪ underaddition.
218
Example 5.61: Let K= 123 456 789 101112159131721 1314152610141822 161718371 192021 222324 252627 282930 aaa aaa aaa aaaaaaaaa aaaaaaaaa aaaaaa aaa aaa aaa aaa LLL LLL LLL LLLLLLLLL LLLLLLLLL , LLLLLL LLL LLL LLL LLL 1151923 4812162024 aaa aaaaaa LLL, LLLLLL 123 456 i 789 101112 aaa aaa aQ{0} aaa aaa LLL LLL LLLLL;1i30 LLL + ∪ ∈≤≤
Example
219
5.62: Let T= () 123 456 7891234567 101112 131415 aaa aaa aaaaaaaaaa aaa aaa LLL LLL LLL,LLLLLLL, LLL LLL 123456 789101112 131415161718i 192021222324 252627282930 aaaaaa aaaaaa aaaaaaaQ{0} aaaaaa aaaaaa LLLLLL LLLLLL LLLLLLLL;1i30 LLLLLL LLLLLL + ∪ ∈≤≤ beasemigroupsupermatrixsemivectorspaceofrefinedlabels overS= R{0} L + ∪ .Sasemigroupundermultiplication. Wecandefinesubstructurewhichisleftasanexerciseto thereader. Howeverwegivesomeexamplesofthem.
Example 5.63: Let 123 456 789 101112 131415123456 161718789101112 192021 222324 252627 282930 aaa aaa aaa aaa aaaaaaaaa aaaaaaaaa aaa aaa aaa aaa LLL LLL LLL LLL LLLLLLLLL V,, LLLLLLLLL LLL LLL LLL LLL =
220
LLLLLL LLLLLL LLLL; LL,LLLLLL1i30 LLLLLL LLLLLL + ∪ ∈ ≤≤ beasemigroupsupermatrixsemivectorspaceofrefinedlabels
+ ∪
W=13
LLLLLLLLLL;
+ ∪ ∈ ≤≤ ⊆
B=
aaa LLL LLL LLL LLL
LLL LLL LLL LLL + ∪ ∈
⊆ V isasemigroupsupermatrixsemivectorsubspaceofrefined
Clearly B ∩ W=
LL LLLL;1i4 + ∪ ∈≤≤⊆ V
123456 789101112 13i 131415161718 24 192021222324 252627282930 aaaaaa aaaaaa aaaQ{0} aaaaaa aa aaaaaa aaaaaa
overthesemigroupS=Q{0} L
.Take
123456i 78910111224 aaaaaaaaaR{0} aaaaaaaa
LLLLLL,LL1i12
V, Wisasemigroupsupermatrixsemivectorsubspaceofrefined labelsoverthesemigroupS=Q{0} L + ∪ .Take
123 456 789 101112 13141513i 16171824 192021 222324 252627 282930 aaa aaa aaa aaa aaaaaaR{0} aaaaa aaa aaa aaa
LLLLLLL; LLL,LL1i30
≤≤
labelsoverthesemigroupS.
13 i 24 aa aR{0} aa
isagainasemigroupsupermatrixsemivectorsubspaceof refinedlabelsoverthesemigroupS.
221
Example 5.64: Let 123 456123456 789789101112 101112131415161718 131415 aaa aaaaaaaaa aaaaaaaaa aaaaaaaaa aaa LLL LLLLLLLLL LLL M,LLLLLL, LLLLLLLLL LLL = 1234 5678 i 9101112 13141516 aaaa aaaa aQ{0} aaaa aaaa LLLL LLLL LLLLLL;1i18 LLLL + ∪ ∈≤≤ beasemigroupsupermatrixsemivectorspaceofrefinedlabels overthesemigroupS=Q{0} L + ∪ undermultiplication.Consider W1= 123 456 789i 101112 131415 aaa aaa aaaaQ{0} aaa aaa LLL LLL LLLLL;1i15 LLL LLL + ∪ ∈≤≤⊆ V, W2= 123456 i 789101112 131415161718 aaaaaa aQ{0} aaaaaa aaaaaa LLLLLL LL; LLLLLL 1i18 LLLLLL + ∪ ∈ ≤≤ ⊆ V andW3= 1234 5678 i 9101112 13141516 aaaa aaaa aQ{0} aaaa aaaa LLLL LLLL LLLLLL;1i16 LLLL + ∪ ∈≤≤
222 ⊆ VbesemigroupsupermatrixsemivectorsubspacesofVof
3
W =
+ ∪
5.65: Let M= 12 34111111 56222222 78333333 910444444 1112555555 1314 aa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa aa LL LLLLLLLL LLLLLLLL LLLLLLLL ,, LLLLLLLL LLLLLLLL LL ()} 12121212i aaaaaaaaaR{0} LLLLLLLLLL;1i14 + ∪ ∈≤≤
L + ∪ .
P1= 12 34111111 56 78 910 1112555555 1314 aa aaaaaaaa aa aa aa aaaaaaaa aa LL LLLLLLLL LL000000 LL000000 , LL000000 LLLLLLLL LL } i aR{0} LL;1i14 + ∪ ∈≤≤⊆ M,
refinedlabelsoverS.ClearlyWi ∩ Wj= φ ifi ≠ j,1 ≤ i,j ≤ 3. AlsoV=
i i1
.Thusisadirectunionofsemigroupsuper matrixsemivectorsubspacesofrefinedlabelsoverR{0} L
=S. Example
beasemigroupsupermatrixsemivectorspaceofrefinedlabels overthesemigroupS=Q{0}
Consider
223
⊆
LLLLLL 000000
aaaaaa LLLLLL
P2= 111111 222222 333333i 444444 555555 aaaaaa aaaaaa aaaaaaaR{0} aaaaaa aaaaaa LLLLLL LLLLLL LLLLLLLL;1i5 LLLLLL LLLLLL + ∪ ∈≤≤
M and P3= (){ 12121212 aaaaaaaa LLLLLLLL, 111111 i aaaaaa aR{0}
000000LL;1i2 000000 000000 + ∪ ∈≤≤⊆ M besemigroupsupermatrixsemivectorsubspacesofM. Wesee P1 ∩ P2= 111111 i 555555 aaaaaa aR{0}
000000 000000LL;i1,5 000000 LLLLLL + ∪ ∈=⊆ M isagainasemigroupsupermatrixsemivectorsubspaceofM.
224 P2 ∩ P3= 111111 1 aaaaaa aR{0} LLLLLL 000000 000000LL 000000 000000 + ∪ ∈⊆ M and P3 ∩ P1= 111111 1 aaaaaa aR{0} LLLLLL 000000 000000LL 000000 000000 + ∪ ∈⊆ M areagainsemigroupsupermatrixsemivectorsubspacesofM overthesemigroupS=Q{0} L + ∪ . WeseethusPi ∩ Pj ≠φ;ifi ≠ j;1 ≤ i,j ≤ 3.ButM= 3 i i1 P = soMisapseudodirectunionofsemigroupsupermatrix semivectorsubspacesofMoverS. Example 5.66: Let P= () 123 456 789 10111212345678 131415 161718 192021 aaa aaa aaa aaaaaaaaaaa aaa aaa aaa LLL LLL LLL LLL,LLLLLLLL, LLL LLL LLL
225 1234567 1234567 1234567i 1234576 7324561 aaaaaaa aaaaaaa aaaaaaaaR{0} aaaaaaa aaaaaaa LLLLLLL LLLLLLL LLLLLLLLL;1i21 LLLLLLL LLLLLLL + ∪ ∈≤≤ beasemigroupofsupermatrixsemivectorspaceofrefined labelsover R{0} L + ∪ =S. Consider M= () 123 456 789 10111211111111 131415 161718 192021 aaa aaa aaa aaaaaaaaaaa aaa aaa aaa LLL LLL LLL LLL,LLLLLLLL LLL LLL LLL } i aR{0} LL + ∪ ∈⊆ PandT={Q{0}
+ ∪ }
L
⊆ Sbeasubsemigroupof S. WeseeMisasemigroupsupermatrixsemivectorspaceof refinedlabelsoverTofPcalledthesubsemigroupsupermatrix semivectorsubspaceofrefinedlabelsoverthesubsemigroupT ofS. Nowwecangivetherelateddefinition. DEFINITION 5.8: Let V be a semigroup super matrix semivector space of refined labels over the semigroup S. Let W (⊆ V) and P (⊆ S) be subsets of V and S respectively, if P be a subsemigroup of S and W is a semigroup super matrix semivector space over the semigroup P of refined labels then we
define W to be a subsemigroup of super matrix semivector subspace of V refined labels over the subsemigroup P of the semigroup S. If V has no subsemigroup super matrix semivector subspace then we define V to be a pseudo simple semigroup super matrix semivector space of refined labels over the semigroup S.
226
5.67: Let V= () 123 456 789 12345678910 101112 131415 161718 aaa aaa aaa aaaaaaaaaa aaa aaa aaa LLL LLL LLL ,LLLLLLLLLL, LLL LLL LLL 123123 312231 123321 i 231132 121233 223231 aaaaaa aaaaaa aaaaaa aR{0} aaaaaa aaaaaa aaaaaa LLLLLL LLLLLL LLLLLL LL;1i18 LLLLLL LLLLLL LLLLLL + ∪ ∈≤≤ beasemigroupsupermatrixsemivectorspaceofrefinedlabels overthesemigroupS=Q{0} L + ∪ .
Interestedreadercansupplyexamplesofthem. Wecandefinelineartransformationofsemigroupsupermatrix semivectorspacesofrefinedlabelsprovidedtheyaredefined overthesamesemigroup.Thistaskislettothereader,however wegivesomeexamplesofthem. Example
227 W= 123456 789101112 131415161718
aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa LLLLLL LLLLLL LLLLLL LLLLLL, LLLLLL LLLLLL 123456789 101112131415161718 aaaaaaaaa aaaaaaaaa LLLLLLLLL LLLLLLLLL, 12 34 56i 78 910 aa aa aaaR{0} aa aa LL LL LLLL;1i36 LL LL + ∪ ∈≤≤ beasemigroupsupermatrixsemivectorspaceofrefinedlabels overthesamesemigroupS=Q{0} L + ∪ . DefineT:V → Wasfollows: T 123 456 789 101112 131415 161718 aaa aaa aaa aaa aaa aaa LLL LLL LLL LLL LLL LLL =
192021222324 252627282930 313233343536
228
LL
LL LL
LLLLLL
LLLLLL LLLLLL LLLLLL
ItiseasilyverifiedTisasemigroupsupermatrix semivectorspacelineartransformationofrefinedlabelsoverthe semigroupS.
123456789 101112131415161718 aaaaaaaaa aaaaaaaaa LLLLLLLLL LLLLLLLLL, T () () 12345678910 aaaaaaaaaa LLLLLLLLLL= 12 34 56 78 910 aa aa aa aa aa
LL LL
and T 123123 312231 123321 231132 121233 223231 aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa LLLLLL LLLLLL LLLLLL
LLLLLL LLLLLL ) = 123123 331312 122331 231122 113311 332132 aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa aaaaaa LLLLLL LLLLLL LLLLLL
.
229
LLL LLL LLL LLL LLLLLLLLLLLL
LLL LLL LLL
beasemigroupsupermatrixsemivectorspaceofrefinedlabels
LLL
LLL LLL
Example 5.68: Let V= 123 456 789 101112 131415123456789 161718101112131415 192021 222324 252627 282930 aaa aaa aaa aaa aaaaaaaaaaaa aaaaaaaaa aaa aaa aaa aaa
, LLLLLLLLLL
LLL 161718 aaa , LL 1234 5678 i 9101112 13141516 aaaa aaaa aR{0} aaaa aaaa LLLL LLLL LLLLLL;1i30 LLLL + ∪ ∈≤≤
overthesemigroupS=Q{0} L + ∪ .DefineamapT:V → V T 123 456 789 101112 131415 161718 192021 222324 252627 282930 aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa
LLL LLL LLL
LLL LLL LLL LLL = 1357 9111315 17192123 25272930 aaaa aaaa aaaa aaaa LLLL LLLL LLLL LLLL ,
230 T123456789 101112131415161718 aaaaaaaaa aaaaaaaaa LLLLLLLLL LLLLLLLLL = 13 2106 3119 41212 51315 61418 715 8161 9172 183 aa aaa aaa aaa aaa aaa aa aaa aaa aa L0L LLL LLL LLL LLL LLL LL0 LLL LLL 0LL and T 1234 5678 9101112 13141516 aaaa aaaa aaaa aaaa LLLL LLLL LLLL LLLL = 12345678 910111213141516 aaaaaaaa aaaaaaaa LLLLLLLL0 0LLLLLLLL . ItiseasilyverifiedTisasemigroupsupermatrix semivectorspacelinearoperatoronV. Nowweproceedontogiveexamplesofthenotionof integersemigroupsupermatrixsemivectorspaceofrefined labels.
231 Example 5.69: Let V= 123 456 1234567 789 891011121314 101112 131415 aaa aaa aaaaaaa aaa aaaaaaa aaa aaa LLL LLL LLLLLLL LLL,, LLLLLLL LLL LLL 1 2 3 4 5 6i 7 8 9 10 11 a a a a a aaR{0} a a a a a L L L L L LLL;1i14 L L L L L + ∪ ∈≤≤ beanintegergroupsupermatrixsemivectorspaceofrefined labelsoverthesemigroupS=Z+ ∪ {0}.
232
Example 5.70: Let V= 112 213 314 41512345 516678910 6171112131415 7181617181920 8192122 920 1021 1122 aa aa aa aaaaaaa aaaaaaa aaaaaaa aaaaaaa aaaa aa aa aa LL LL LL LLLLLLL LLLLLLL LLLLLLL , LLLLLLL LLLLL LL LL LL 232425 aaa , LL 12 34i 56 aa aaaR{0} aa LL LLLL;1i25 LL + ∪ ∈≤≤ beanintegersemigroupsupermatrixsemivectorspaceof refinedlabelsoverthesemigroup3Z+ ∪ {0}. Substructures,lineartransformationcanbedefinedasin caseofusualsemigroupsupermatrixsemivectorspacesof refinedlabels. Nowweproceedontodefinethenotionofgroupsupermatrix semivectorspacesofrefinedlabelsoveragroup.
DEFINITION 5.9: Let V be a semigroup super matrix semivector space of refined over the semigroup S = Q L + ; ( R L + ) under multiplication. If S is a group under multiplication then we
233 define V to be a group super matrix semivector space of refined labels over the multiplication group Q L + or R L + . Wewillfirstillustratethissituationbysomeexamples. Example 5.71: Let V= 1234567 28910111213 12391415161718 344101519202122 565111620232425 6121721242627 7131822 aaaaaaa aaaaaaa aaaaaaaaa aaaaaaaaa aaaaaaaaa aaaaaaa aaaa LLLLLLL LLLLLLL LLLLLLLLL LL,LLLLLLL LLLLLLLLL LLLLLLL LLLLL 252728 aaa , LL } 12345678i aaaaaaaaaR{0} LLLLLLLLLL;1i28 + ∪ ∈≤≤ beagroupsupermatrixsemivectorspaceofrefinedlabelsover themultiplicativegroupQ L + Example 5.70: Let M= 112 213 314 415 5161234567 617891011121314 718 819 920 1021 aa aa aa aa aaaaaaaaa aaaaaaaaa aa aa aa aa LL LL LL LL LLLLLLLLL ,, LLLLLLLLL LL LL LL LL
234
LLLLLLLLL LLLLLLLLL LLLLLLLLL
LLLLLLLL
}
() 123 456
789
aaa
aaa
LLL
LLL
= 1212345
34678910 aaaaaaa
aaaaaaa
123456789 101112131415161718 192021222324252627 282930313233343536 373839404142434445 46474849505152 aaaaaaaaa aaaaaaaaa aaaaaaaaa aaaaaaaaa aaaaaaaaa aaaaaaa
LLLLLLLLL LLLLLLLLL
5354aa L
i aR{0} LL;1i54 + ∪ ∈≤≤ beagroupsupermatrixsemivectorspaceofrefinedlabelsover R L + thegroupofpositiverefinedlabels. Wecandefineasamatterofroutinethesubstructuresinthem. Herewepresentthembysomeexamples. Example 5.73: Let
12345678910
101112 aaa
aaaaaaaaaa
aaa
LLL V,LLLLLLLLLL,
LLL
i
aR{0}
LLLLLLL ,LL;1i12 LLLLLLL + ∪ ∈≤≤ beagroupsupermatrixsemivectorspaceofrefinedlabelsover thegroupG=Q L + . Nowconsider
235 P= 123 45612
78934
aaa aaaaa
LLL LLLLL
LLLLL LLL + ∪
⊆ Visagroupsupermatrixsemivectorsubspaceofrefined
+
Let V= 12 34 56 78
LL LL LL
LL LL 12345678 910111213141516 aaaaaaaa
LLLLLLLL LLLLLLLL,
27891011 3812131415i 4913161718 51014171920 61115182021
aaaaaaaR{0}
aaaaaa aaaaaa LLLLLL
LLLLLL LLLLLL
∪
i
101112
aR{0} aaaaa aaa
,LL;1i12
∈≤≤
labelsoverthegroupG=Q L
undermultiplication. Example 5.72:
910 aa aa aa aa aa
,
aaaaaaaa
123456
aaaaaa aaaaaa
aaaaaa
LLLLLL LLLLLLLL; LLLLLL1i21
+
∈ ≤≤ beagroupsupermatrixsemivectorspaceofrefinedlabelsover thegroupG=Q L + ,Q L + groupundermultiplication.
236 Consider P1= 12 34 56i 78 910 aa aa aaaR{0} aa aa LL LL LLLL;1i10 LL LL + ∪ ∈≤≤⊆ V, P2= 12345678i 910111213141516 aaaaaaaaaR{0} aaaaaaaa LLLLLLLLLL; LLLLLLLL1i16 + ∪ ∈ ≤≤ ⊆ V and P3= 123456 27891011 3812131415i 4913161718 51014171920 61115182021 aaaaaa aaaaaa aaaaaaaR{0} aaaaaa aaaaaa aaaaaa LLLLLL LLLLLL LLLLLLLL; LLLLLL1i21 LLLLLL LLLLLL + ∪ ∈ ≤≤ ⊆ V begroupsupermatrixsemivectorsubspacesofrefinedlabelsof VoverthemultiplicativegroupQ L + . Wesee V= 3 i i1 P = ;Pi ∩ Pj= φ ifi ≠ j,1 ≤ i,j ≤ 3, thusVisadirectsumofgroupsupermatrixsemivector subspacesofrefinedlabelsofV.
237
LLLLL
() 12 34 56 78 9101234567 1112 1314 1516 1718 aa aa aa aa aaaaaaaaa aa aa aa aa LL LL LL LL LL,LLLLLLL LL LL LL LL } i aR{0} LL;1i20 + ∪ ∈≤≤ beagroupsupermatrixsemivector spaceofrefinedlabelsoverthegroupG=Q L + ,Q L + =Gisa multiplicativegroup. Consider
Example 5.75: Let V= 1234 125678 349101112 13141516 aaaa aaaaaa aaaaaa aaaa LLLL LLLLLL ,,LLLLLL LLLL 12345 678910 1112131415 1617181920 aaaaa aaaaa aaaaa aaaaa LLLLL LLLLL
LLLLL
238
H1= () 12i 1234567 34 aaaR{0} aaaaaaa aa LLLL; LL,LLLLLLL1i7 + ∪ ∈ ≤≤ ⊆ V, H2= 1234 5678 9101112 13141516 aaaa aaaa aaaa aaaa LLLL LLLL LLLL, LLLL () 1234567 aaaaaaa LLLLLLL| } i aR{0} LL;1i16 + ∪ ∈≤≤⊆ V, H3= () 12345 678910 123 1112131415 1617181920 aaaaa aaaaa aaa aaaaa aaaaa LLLLL LLLLL LL0000L, LLLLL LLLLL } i aR{0} LL;1i20 + ∪ ∈≤≤⊆ V, H4= () 12 34 56 78 i 910123457 1112 1314 1516 1718 aa aa aa aa aR{0} aaaaaaaa aa aa aa aa LL LL LL LL LLLL; ,LLLLL0L LL1i18 LL LL LL + ∪ ∈ ≤≤ and
239 H5= () {} 1234567i aaaaaaaaR{0} LLLLLLLLL;1i7 + ∪ ∈≤≤⊆ Vbegroupsupermatrixsemivectorsubspacesofrefinedlabels ofVoverthegroupQ L + . Wesee=Hi ∩ Hj= φ ifi ≠ j,1 ≤ i,j ≤
5 i i1 H = ;thusVisthepseudodirectunionofgroupsupermatrix semivectorsubspacesofrefinedlabelsoverthegroupQ
+
WofrefinedlabelslineartransformationsprovidedVandW
Wewillillustratethissituationbysomeexamples. Example 5.76: Let V= () 1 2 3 4 5 12 1234567896 34 7 8 9 10 11 a a a a a aa aaaaaaaaaa aa a a a a a L L L L L LL L LL,LLLLLLLLL,, L L L L L
5.FurtherV=
L
=G. WecandefinegroupsupermatrixsemivectorspacesVand
aredefinedoverthesamegroupG.Wecanalsodefine subgroupsupermatrixsemivectorsubspaceofrefinedlabelsof VoverthesubgroupofG.
beagroupsuperrowsemivectorspaceofrefinedlabelsoverthe
WeseeWisagroupsupermatrixsemivectorspaceofrefined labelsoverthegroupH,thusWisasubgroupsupermatrix semivectorsubspaceofrefinedlabelsoverthesubgroupHof thegroupG.
IfGhasnopropersubgroupsthenwedefineVtobea pseudosimplegroupvectorspaceofrefinedlabelsoverG.
LikewiseifVthegroupsupermatrixsemivectorspaceof refinedlabelsoverthegroupGhasasubspaceSwhichisovera semigroupH ⊆ GwecallSapseudosemigroupsupermatrix semivectorsubspaceofVoverthesemigroupHofthegroupG.
Wewilljustillustratethissituationbysomeexamples.
240
12345 12345 12345 12345i 12345 12345 12345 aaaaa aaaaa aaaaa aaaaaaR{0} aaaaa aaaaa aaaaa LLLLL LLLLL LLLLL LLLLLLL;1i11 LLLLL LLLLL LLLLL + ∪ ∈≤≤
groupG=R L + undermultiplication. Consider W= () 12 123456789 34 aa aaaaaaaaa aa LL LL,LLLLLLLLL } i aR{0} LL;1i9 + ∪ ∈≤≤⊆ V;considerH=Q L +⊆ R L + =G,Q L + =HisasubgroupofGundermultiplication.
241 Example 5.77: Let V= () 1 2 3 4 5 12345678 6 7 8 9 10 a a a a a aaaaaaaa a a a a a L L L L L L,LLLLLLLL, L L L L 12345613 78910111214 15161718192021i 22232425262728 29303132333435 aaaaaaa aaaaaaa aaaaaaaaR{0} aaaaaaa aaaaaaa LLLLLLL LLLLLLL LLLLLLLLL;1i35 LLLLLLL LLLLLLL + ∪ ∈≤≤ beagroupsupermatrixsemivectorspaceofrefinedlabelsover thegroupG=Q L + undermultiplication. Take P= (){ 12345678 aaaaaaaa LLLLLLLL
L L L L L L L L L L
a a a a a a a a a a
1 2 3 4 5 6 7 8 9 10
} i aR{0} LL;1i10 + ∪ ∈≤≤⊆ V
beapseudosemigroupsupermatrixsemivectorsubspaceof refinedlabelsofVoverthesemigroup S= n 1 2 Ln0,1,2,...,=∞⊆ Q L + undermultiplication.
ItisinterestingtonotethatthisVhasinfinitelymanypseudo semigroupsupermatrixsemivectorsubspacesofrefinedlabels.
Thefollowingtheoremguaranteestheexistenceofpseudo semigroupsupermatrixsemivectorsubspacesofrefinedlabels. RecallagroupGissaidtobeaSmarandachedefinitegroup ifithasapropersubsetH ⊂ GsuchthatHisthesemigroup undertheoperationsofG.WeseeQ L + and R L + are Smarandachedefinitegroupsastheyhaveinfinitenumberof semigroupsundermultiplications.
THEOREM 5.1: Let V be a group super matrix semivector space of refined labels over the group G = Q L + (or R L + ). V has infinitely many pseudo semigroup super matrix semivector
242
243 subspaces of refined labels of V over subsemigroups
multiplication of the group G = Q L +
R L + ). Interestedreadercanderiveseveralrelatedresults.Howeverwe candefinepseudosetsupermatrixsemivectorsubspaceof refinedlabelsofagroupsupermatrixsemivectorspaceof refinedlabelsoverthemultiplicativegroupQ L + and R L + . Wewillonlyillustratethissituationbyanexampleortwo. Example 5.78: Let V= 123 456 789 aaa aaa aaa LLL LLL, LLL 12345678 910111213141516 12345678 aaaaaaaa aaaaaaaa aaaaaaaa LLLLLLLL LLLLLLLL LLLLLLLL 123456 789101112 123456 i 789101112 123456 789101112 aaaaaa aaaaaa aaaaaa aR{0} aaaaaa aaaaaa aaaaaa LLLLLL LLLLLL LLLLLL LL;1i16 LLLLLL LLLLLL LLLLLL + ∪ ∈≤≤ beagroupsupermatrixsemivectorspaceofrefinedlabelsover themultiplicativegroupG=Q L + . Consider P= nn 11 25 LLn0,1,2,...∪=⊆ Q L + ;
H under
(or
PisonlyasubsetofQ L + =G.
Take M= 1231467 4569121415 78914
aaaaaaa aaaaaaa aaaaa
LLLL00L0LL0 LLL,L00L0LL0 LLLL00L0000 } i aR{0} LL;i1,2,...,9,12,14,15 + ∪ ∈=⊆ V.
Misasetsupermatrixsemivectorspaceofrefinedlabels overthesetP ⊆ G.ThusMisapseudosetsupermatrix semivectorspaceofrefinedlabelsofVoverthesetPofthe multiplicativegroupG.
Thispseudosetsupermatrixsemivectorspaceofrefined labelscanalsobedefinedincaseofsemigroupsupermatrix semivectorspaceofrefinedlabelsoverthesemigroupQ{0} L + ∪ or R{0} L + ∪ .
Thetaskofstudyingthisnotionandgivingexamplestothis effectisleftasanexercisetothereader.
244
Applicationsofsupermatrixvectorspacesofrefinedlabels andsupermatrixsemivectorspaceofrefinedlabelsisatavery dormantstate,asonlyinthisbooksuchconceptshavebeen definedanddescribed.Thestudyofrefinedlabelsisvery recent.Theintroductionofsupermatrixvectorspacesof refinedlabels(ordinarylabels)areverynew.Certainlythese newnotionswillfindapplicationsinthesuperfuzzymodels, qualitativebelieffunctionmodelsandotherplacesweresuper matrixmodelcanbeadopted.
245
SinceDSmareappliedinseveralfieldstheinterested researchercanfindapplicationinappropriatemodelswhere DSmfindsitsapplications.Whenoneneedsthebulkworkto savetimesupermatrixofrefinedlabelscanbeusedfor informationretrieval,fusionandmanagement.
Furtherasthestudyisverynewtheapplicationsofthese structureswillbedevelopedbyresearchesinduecourseoftime.
246
Inthischapterwehavesuggestedoverhundredproblemssome ofwhichareopenresearchproblems,someoftheproblemsare simple,mainlygiventothereaderforthebetterunderstanding ofthedefinitionsandresultsaboutthealgebraicstructureofthe refinedlabels.Wesuggestproblemswhicharebothinnovative andinteresting.
1.ObtainsomeinterestingpropertiesaboutDSmsuperrow vectorspaceofrefinedlabelsoverLQ.
2.ObtainsomeinterestingpropertiesaboutDSmsuper columnvectorspaceofrefinedlabelsoverLR.
3.FinddifferencesbetweentheDSmsupervectorspacesof refinedlabelsdefinedoverLRandLQ.
247
248
1 2 3 4 i 5 6 7 8 a a a a aR{0} a a a a L L L L LL;1i8 L L L L + ∪ ∈≤≤
4.LetVbeasuperrowvectorspacegivenby; V= 14710 25811i 36912 aaaa aaaaaR aaaa LLLL LLLLLL;1i12 LLLL ∈≤≤ ; a)FindabasisofVoverLR. b)WhatisthedimensionofVoverLR? c)GiveasubsetofVwhichislinearlydependent. 5.LetV= () {} 12345i aaaaaaR LLLLLLL;1i5 ∈≤≤ be asuperrowvectorspaceofrefinedlabelsoverthefieldLR. a)FindanoninvertiblelinearoperatoronV. b)FindaninvertiblelinearoperatoronV. c)FindthealgebraicstructureenjoyedbyR L L(V,V) d)DoesT:V → VbesuchthatkerTisahypersubspace ofV? 6.LetV=
beasupercolumn vectorspaceofrefinedlabelsoverLR. Studyquestions(a)to(d)describedinproblem(5).
7.Obtainsomeinterestingpropertiesaboutsuperrowmatrix vectorspacesofrefinedlabels.
LetV=
L L L L L LLL;1i11 L L L L L
1 2 3 4 5 6i 7 8 9 10 11
a a a a a aaR a a a a a
∈≤≤ beasupercolumn
vectorspaceofrefinedlabelsoverLR.
a)WriteVasadirectsumofsupercolumnvector subspacesofVoverLR.
b)WriteVasapseudodirectsumofsupercolumnvector subspacesofVoverLR.
c)DefinealinearoperatoronVwhichisaprojection.
d)FinddimensionofVoverLR.
e)FindabasisofVoverLR.
249
250 8.Let W= 12345 678910 1112131415
2122232425
3132333435 aaaaa aaaaa aaaaa aaaaaaR aaaaa aaaaa aaaaa LLLLL LLLLL LLLLL LLLLLLL;1i35 LLLLL LLLLL LLLLL ∈≤≤ beasupermatrixvectorspaceofrefinedlabelsoverthe fieldLR. a)FindsubspacesWiinVsothatV= 5 i i1 W = isadirect sum. b)FindabasisforVoverLR. c)FinddimensionofVoverLR. d)SupposeM= 1 2 3412 5613i 7814 91015 11 a a aaa aaaaR aaa aaa a 00L00 00L00 LL0L0 LL00LLL;1i15 LL0L0 LL00L 00L00 ∈≤≤⊆ WbeasupermatrixvectorsubspaceofVoverLRfind T:W → WsuchthatT(W) ⊆ W. e)IsTinvertible? f)FindanoninvertibleTonW. g)WriteVasapseudodirectsum.
1617181920i
2627282930
9.LetP=
LLLL LLLL LLLLLL;1i16 LLLL
1234 5678 i 9101112 13141516
aaaa aaaa aR aaaa aaaa
∈≤≤ bea
supermatrixvectorspaceofrefinedlabelsoverLR. a)FindabasisforPoverLR. b)IfLRisreplacedbyLQwhatwillbethedimensionofP? c)WritePasapseudodirectsumofsupervector subspacesofrefinedlabelsoverLR. 10.LetM=
LLLL LLLL LLLLLL;1i20 LLLL LLLL
1234 5678 9101112i 13141516 17181920
aaaa aaaa aaaaaR aaaa aaaa
∈≤≤ be asupermatrixvectorspaceovertherefinedlabeledfield LR. a)ProveMisinfinitedimensionaloverLQ. b)WriteMasadirectunionofsupermatrixvectorspaces. c)FindS=R L L(M,M);whatisthealgebraicstructure enjoyedbyS. 11.LetV= 123 456i 789
aaa aaaaR aaa
LLL LLLLL;1i9 LLL ∈≤≤ beasuper matrixvectorspaceoverLRofrefinedlabels.
251
252
M= 12 34 56 78i 910 1112 1314 aa aa aa aaaR aa aa aa LL LL LL LLLL;1i14 LL LL LL ∈≤≤ beasupermatrix vectorspaceofrefinedlabelsoverLR. a)DefineaT:V → MsothatkerTisanontrivial subspaceofV. b)Definep:V → Msuchthatkerp= 000 000 000 . c)FindthealgebraicstructureenjoyedbyS = R L Hom(V,M)=R L L(V,M). d)WhatisthedimensionofMoverLR? e)Find η :M → Vsothatker η≠ 00 00 00 00 00 00 00 . f)FindB=R L Hom(M,V)=R L L(M,V). g)CompareBandS.
253 12.Let M= 12345 678910 1112131415 i 1617181920 2122232425 2627282930 aaaaa aaaaa aaaaa aR aaaaa aaaaa aaaaa LLLLL LLLLL LLLLL LL;1i30 LLLLL LLLLL LLLLL ∈≤≤ beasupermatrixvectorspaceoverLR. a)Let P= 12 345 i 67 89 1011 aa aaa aR aa aa aa 000LL 00000 LLL00 LL;1i11 000LL 000LL 000LL ∈≤≤⊆ M.FindT:M → MsothatT(P) ⊆ P. find η :M → Msuchthat η (P) ⊄ P. b)LetV=1357i 2468 aaaa aR aaaa LLLL LL;1i8 LLLL ∈≤≤ be asuperrowvector,vectorspaceofrefinedlabelsover LR. i)Definealinearfunctionf:V → LR. ii)Iskerfasuperhyperspace? 13.Obtainsomeinterestingresultsonsupermatrixvectorspace ofrefinedlabelsoverLR. 14.LetV=LRbeavectorspace(linearalgebra)overLQ. ProvedimensionofVisinfiniteoverLQ.
254
20.Let V= 123417 567818 i 910111219 1314151620 aaaaa aaaaa aR{0} aaaaa aaaaa LLLLL LLLLL LL;1i20 LLLLL LLLLL + ∪ ∈≤≤ 21.beasuperrowvectorsemivectorspaceofrefinedlabels over R{0} L + ∪ . a)FindabasisforVoverR{0} L + ∪ .
15.ProveV=LRisfinitedimensionaloverLR. 16.LetV= 123 456i 789 aaa aaaaR aaa LLL LLLLL;1i9 LLL ∈≤≤ beasuper matrixvectorspaceofrefinedlabelsoverLR.ProveVis notasupermatrixlinearalgebraofrefinedlabelsoverLR. 17.ProveasupermatrixvectorspaceofrefinedlabelsoverLR canneverbeasupermatrixlinearalgebraofrefinedlabels overLR. 18.LetM= () {} 12345i aaaaaaR LLLLLLL;1i5 ∈≤≤ be asuperrowvectorspaceofrefinedlabelsoverLR.How manysuchsuperrowvectorspacesofrefinedlabelscanbe constructedusing1 × 5superrowvectorsbyvaryingthe partitionon () 12345 aaaaa LLLLL? 19.Provethoseclassicaltheoremswhichcanbeadoptedon supermatrixvectorspaceofrefinedlabels.
255
LLLL LLLL
→
e)Finda η :V →
f)Can η :V → Vbeonetooneyet η -1cannotexist?
b)FindasetoflinearlyindependentelementsofVwhich isnotabasisofVover R{0} L + ∪ . c)FindalinearlydependentsubsetofVoverthesemifield R{0} L + ∪ . d)ProveVhaslinearlyindependentsubsetswhose cardinalityisgreaterthanthatofthebasis. 21.LetV= 1234 5678 9101112 13141516 i 17181920 21222324 25262728 29303132 aaaa aaaa aaaa aaaa aR{0} aaaa aaaa aaaa aaaa LLLL
LLLL LL;1i32 LLLL LLLL LLLL LLLL + ∪ ∈≤≤ besupercolumnvector,semivectorspaceofrefinedlabels overthesemifield R{0} L + ∪ . a)Findtwosupercolumnvectorsemivectorsubspacesof Vwhichhaszerointersectionover R{0} L + ∪ . b)WriteVasadirectsumofsupercolumnvector semivectorsubspacesoverR{0} L + ∪ . c)WriteVasapseudodirectsumofsupercolumnvector semivectorsubspacesoverR{0} L + ∪ . d)FindaT:V
Wwhichhasanontrivialkernel.
Vsothatthe η isaninvertiblelinear transformationonV.
Justify.
256
L
22.LetV= 1 2 3 4 5 6 7i 8 9 10 11 12 13 a a a a a a aaQ{0} a a a a a a
L L L L L LLL;1i13 L L L L L L + ∪ ∈≤≤ beasupercolumn semivectorspaceofrefinedlabelsoverQ{0} L + ∪ . a)FindabasisofVoverQ{0} L + ∪ . b)CanVhavemorethanonebasis? c)FinddimensionofVoverQ{0} L + ∪ . d)WriteVasapseudodirectsumofsupercolumn semivectorsubspacesoverQ{0} L + ∪ .
257 e)SupposeW= 1 2 3 4i 5 6 7 a a a aaQ{0} a a a L 0 L 0 L 0 LLL;1i7 0 L 0 L 0 L + ∪ ∈≤≤⊆ Vbea supercolumnsemivectorsubspaceofVofrefined labelsover Q{0} L + ∪ . i)FindaT:V → VsuchthatT(W) ⊆ W. ii)FindT 1:V → VsuchthatT 1(W) ⊄ W. f)FindasubsetofVwithmorethan14elementswhichis alinearlyindependentsubsetofVoverQ{0} L + ∪ 23.LetV= 1234 5678 9101112 i 13141516 17181920 21222324 aaaa aaaa aaaa aR{0} aaaa aaaa aaaa LLLL LLLL LLLL LLLLLL;1i24 LLLL LLLL + ∪ ∈≤≤ beasupermatrixsemivectorspaceofrefinedlabelsover Q{0} L + ∪ .
258
+ ∪ .
→
+ ∪
LLLLL LLLLL LLLLLLL;1i20 LLLLL
∪ ∈≤≤
∪
123
789
131415i 161718 192021 222324 252627
aaaaR{0} aaa aaa aaa aaa LLL LLL LLL LLL LLLLL;1i27 LLL LLL LLL LLL + ∪ ∈≤≤
a)IsVfinitedimensionaloverQ{0} L + ∪ ? b)CanVhavefinitelinearlyindependentsubset? c)CanVhavefinitelinearlydependentsubsets? d)Find Q{0} L Hom(V,V) + ∪ =S. e)WhatisthealgebraicstructureenjoyedbyS? f)IsSafiniteset? g)GivetwomapsT:V → VandP:V → VsothatTP= PT. h)FindT f:V → Q{0} L
i)IfK={f|f:V
Q{0} L
},whatisthecardinalityof K? j)DoesKhaveanynicealgebraicstructureonit? 24.Let V= 12345 678910 i 1112131415 1617181920 aaaaa aaaaa aQ{0} aaaaa aaaaa
+
beasupermatrixsemivectorspaceofrefinedlabelsover Q{0} L +
}and W=
456
101112
aaa aaa aaa aaa
259 beasupercolumnvectorsemivectorspaceofrefinedlabels over Q{0} L + ∪ . a)DefineT:V →
transformationonV. b)ProveWisaninfinitedimensionalspace. c)If η :V → Wdoes η enjoyanyspecialpropertyrelated
d)CanP:V → WsuchthatkerPisdifferentfromthe zerospace? e)Whatisthealgebraicstructureenjoyedby Q{0} L Hom(V,W) + ∪ ? 25.LetVbeasuperrowvectorspaceofrefinedlabelsoverLR; whereV= () {} 123ni aaaaaR LLL...LLL;1in ∈≤≤ ; a)FindR L Hom(V,V). b)IsR L Hom(V,V)asuperrowvectorspace? c)IsR L Hom(V,V)justavectorspaceofrefinedlabels? Justifyyourclaim. 26.Let V= 12345 678910 1112131415i 1617181920 2122232425 aaaaa aaaaa aaaaaaQ aaaaa aaaaa LLLLL LLLLL LLLLLLL;1i25 LLLLL LLLLL ∈≤≤ beasupermatrixvectorspaceofrefinedlabelsoverthe refinedfieldLQ. a)WhatisthedimensionofVoverLQ? b)DoesVhavesupermatrixrowvectorsubspaceover LQ? c)FindasetlinearlydependentelementsinVoverLQ. d)FindthealgebraicstructureenjoyedbyQ L L(V,V)= Q L Hom(V,V).
WsuchthatTisaninvertiblelinear
withdimensionsofVandW?
260
27.LetV= 1234 5678 9101112 13141516 17181920i 21222324 25262728 29303132 33343536 aaaa aaaa aaaa aaaa aaaaaR aaaa aaaa aaaa aaaa LLLL LLLL LLLL LLLL LLLLLL;1i36 LLLL LLLL LLLL LLLL ∈≤≤ be asupermatrixvectorspaceofrefinedlabelsoverLR. a)FindasupermatrixvectorsubspaceWofVdimension 19ofrefinedlabelsoverLR. b)FindT:V → VsothatT(W) ⊆ W. c)FindP:V → VsothatP(W) ⊄ W. d)ComparethesuperlinearoperatorsPandTofV. e)WhatiskerT? f)FindkerPandcompareitwithkerT. g)Findasupermatrixvectorsubspaceofdim5ofVof refinedlabelsoverLR. h)WriteVasapseudodirectsumofsupermatrixvector subspaces. 28.Let V= 159131721 2610141822i 3711151923 4812162024 aaaaaa aaaaaaaR aaaaaa aaaaaa LLLLLL LLLLLLLL; LLLLLL1i24 LLLLLL ∈ ≤≤ beasupermatrixvectorspaceofrefinedlabelsoverLQ. a)FinddimensionofVoverLQ. b)FindQ L Hom(V,V).
261
29.LetM= 12345 26789 37123i 48945 56356
aaaaaaR aaaaa aaaaa
LLLLL LLLLLLL;1i9 LLLLL LLLLL
P= 1 67 711745 45 56 a aa aaaaaaR aa aa L0000 0LL00 0LL00L,L,L,LL 000LL 000LL ∈⊆ V e)FindalinearoperatorTonVsuchthatT(P) ⊆ P. f)LetM= 2345 289 323234589 4 5 aaaa aaa aaaaaaaaaR a a 0LLLL L00LL L00LLL,L,L,L,L,LL L0000 L0000 ∈
c)WriteV= 6 i i1 W = sothatVisadirectunionofsuper matrixvectorsubspacesofVoverLQ. d)WriteV= 6 i i1 W = sothatVisthepseudodirectunionof supermatrixvectorsubspacesofV.
aaaaa aaaaa
LLLLL
∈≤≤ beasupermatrixvectorspaceofrefinedlabelsoverLR. a)FinddimensionofVoverLR. b)IsdimVoverRlessthan25? c)CanVhaveasupermatrixvectorsubspacesofrefined labelsofdimension9overLR? d)Findthedimensionofthesupermatrixvectorsubspace
262 ⊆ V,beasupermatrixvectorspaceofrefinedlabels
. g)FindT:V
30.LetV= 12345 12345 12345 12345 12345i 12345 12345 12345 12345 aaaaa
aaaaa aaaaa aaaaa aaaaa LLLLL LLLLL LLLLL LLLLL LLLLLLL;1i5 LLLLL LLLLL LLLLL LLLLL ∈≤≤
d)FindP
P
P
2P
overLRofV. FinddimensionofMoverLR
→ VsoT(M) ⊄ M.
aaaaa aaaaa aaaaa aaaaaaR
beasupermatrixvectorspaceofrefinedlabelsoverLR. a)WhatisdimensionofVoverLR? b)IsVofdimension45overLR?Justifyyouranswer. c)DefineT:V → VsothatkerT= 00000 00000 00000 00000 00000 00000 00000 00000 00000 .
1andP2twodistinctlinearoperatorsofVsothat
1
2=P
1=IdentityoperatoronV. 31.Givesomeinterestingapplicationsofsupermatrixvector spacesofrefinedlabelsoverLRorLQ.
263 32.Provewecannotingeneralconstructsupermatrixlinear
RorLQ. 33.Obtainsomeinterestingpropertiesenjoyedbyintegerset supermatrixsemivectorspaceofrefinedlabelsdefinedover thesetS=3Z+ ∪ 5Z+ ∪ {0}. 34.Obtainsomeinterestingpropertiesbygroupsupermatrix semivectorspacesofrefinedlabelsdefinedoverQ L + or R L + . 35.LetV= 1 2 3 123 4 456 5 78912345678 6 1011129101112 7 131415 8 161718 9 10 11 a a a aaa a aaa a aaaaaaaaaaa a aaaaaaa a aaa a aaa a a a L L L LLL L LLL L LLLLLLLLLLL L LLLLLLLL L LLL L LLL L L L 13141516 aaaa LLL i aR{0} LL;1i18 + ∪ ∈≤≤ }beaintegersupermatrixsemivector spaceofrefinedlabelsover3Z+ ∪ 5Z+ ∪ {0}=S. a)FindabasisofVoverS. b)FindintegersupermatrixsemivectorsubspaceofVof refinedlabels. c)WriteV= 3 i i1 W = asadirectsum.
algebrasofrefinedlabelsdefinedoverL
264 36.LetV= 123123123 456456456i 789789789 aaaaaaaaa aaaaaaaaaaQ aaaaaaaaa LLLLLLLLL LLL,LLL,LLLLL;1i9 LLLLLLLLL ∈≤≤ beasetsupermatrixvectorspaceofrefinedlabelsover thesetS= Q{0} L + ∪ ∪ R{0} L + ∪ . a)FindsubspacesofV. b)WriteVasapseudodirectsumofsubspaces. 37.LetV= 1 2 3 4 121234567 5 341234567 6 7 8 9 a a a a aaaaaaaaa a aaaaaaaaa a a a a L L L L LLLLLLLLL L LL,,LLLLLLL L L L L } i aR{0} LL;1i9 + ∪ ∈≤≤ beagroupsupermatrixsemivector spaceofrefinedlabelsover R L + themultiplicativegroup. a)FindV=i i1 W = asapseudodirectsum. b)FindsubgroupsupermatrixsemivectorsubspaceofV ofrefinedlabelsover R L + . c)Findpseudosemigroupsupermatrixsemivector subspacesofVofrefinedlabelsoverR L + .
265
38.LetV= 12 12 34
3434
5656
56
aaaaaaaaa aaaa
LL LL LL LLLL LLLL LL,,LLLLLLLLL LL LLLL LL LL LL } i
∪ ∈≤≤
LL;1i2 LL
d)FindpseudosetsupermatrixsemivectorsubspaceofV ofrefinedlabelsover R L + .
3412
123456789
5678
56 78 aa aa aa aaaa aaaa
aaaa aa aa aa
aR{0} LL;1i9 +
beasemigroupsupermatrixsemivector spaceofrefinedlabelsoverthesemigroupS=Q{0} L + ∪ under addition. a)WriteV=iWasapseudodirectsumofsubspaces. b)Findtwopseudosetsupermatrixsemivectorsubspaces ofrefinedlabelsofVoverS. c)FindT:V → Vwithnontrivialkernel. d)FindP:V → VsothatP -1exists. e)IfK= 12 34 i 56 78 aa aa aQ{0} aa aa LL LL
LL + ∪ ∈≤≤⊆ V;find dimensionofKover Q{0} L + ∪ .
266 39.Obtainsomeniceapplicationsofsemigroupsupermatrix
R{0} L + ∪ underaddition. 40.LetV= 1234 5678 9101112 13141516i 17181920 21222324 25262728 aaaa aaaa aaaa aaaaaR aaaa aaaa aaaa LLLL LLLL LLLL LLLLLL;1i28 LLLL LLLL LLLL ∈≤≤ be asupermatrixvectorspaceofrefinedlabelsoverLR. a)FinddimensionofVoverLR. b)Findsubspacesofdimensions5,7and20. c)WriteV=W
⊕ … ⊕ W
d)FindT:V → VsothatTisanidempotentoperatoron
e)FindT:V
41.Let P= 1 2 31234 45678 12i 59101112 34 613141516 717182019 8 9 a a aaaaa aaaaa aaaR aaaaa aa aaaaa aaaaa a a L L LLLLL LLLLL LLLL; LLLLL LL,,1i19 LLLLL LLLLL L L ∈ ≤≤ beasetsupermatrixvectorspaceofrefinedlabelsovertheset
semivectorspaceofrefinedlabelsoverthesemigroup
1
twhereWi’saresubspacesofV.
V.
→ VsothatT -1exists.
267 S= nn 11 2{0}5{0} LLn0,1,2,..., ∪∪ ∪∈∞⊆ LR. a)WriteV= 3 i i1 W = asadirectsumofsubspaces. b)WriteV= 5 i i1 W = asapseudodirectsumofsubspaces? c)FindtwodisjointsubspacesW1andW2andfindtwo linearoperatorsT 1andT 2sothatT 1(W1) ⊆ W1and T 2(W2) ⊆ W2. d)Findasubsetsupermatrixvectorsubspaceofrefined labelsofV. 42.Writedownanyspecialfeatureenjoyedbythesupermatrix semivectorspaceofrefinedlabelsoverR{0} L + ∪ or Q{0} L + ∪ . 43.Canwehavefinitesupermatrixsemivectorspaceofrefined labelsoverthesemifield R{0} L + ∪ or Q{0} L + ∪ ?Justifyyour claim. 44.Isitpossibletoconstructsetsupermatrixsemivectorspace
Q{0} L + ∪ offiniteorder? 45.LetP= 12345 21534 35412i 43125 54253 aaaaa aaaaa aaaaaaR aaaaa aaaaa LLLLL LLLLL LLLLLLL;1i5 LLLLL LLLLL ∈≤≤ be asupermatrixvectorspaceofrefinedlabelsoverthefield LR. a)FinddimensionofPoverLR. b)WriteP= ∪ Wi
i
c)CanPhavetendimensionalsupermatrixvector
R?Justifyyouranswer.
ofrefinedlabelsover
,W
,subspacesasadirectsum.
subspaceofrefinedlabelsoverL
1),…,f(v6
268
d)DefineT:P → PsothatT 2=T. e)FindS:P → PsothatS2=(0). f)FindhypersubspacesofrefinedlabelsinPoverLR. 46.ObtainsomeinterestingpropertiesaboutRLR L(V,L)where VisasupermatrixvectorspaceofrefinedlabelsoverLR. 47.LetV= 1234 5678 1234 i 5678 1234 5678 aaaa aaaa aaaa aR aaaa aaaa aaaa LLLL LLLL LLLL LL;1i8 LLLL LLLL LLLL ∈≤≤ bea supermatrixvectorspaceofrefinedlabelsoverLR.Define f:V → LRsothatfisalinearfunctionalonV.Supposev1, …,v6aresomesixlinearlyindependentelementsofV; findf(vi);i=1,…,6.Studytheset{f(v
)} ⊆ LR.Whatisthealgebraicstructureenjoyedby R LR L(V,L)?Howmanyhypersubspacesexistincaseof V? 48.LetV= 1234 2341 i 3421 4133 aaaa aaaa aR{0} aaaa aaaa LLLL LLLL LLLLLL;1i4 LLLL + ∪ ∈≤≤ be aintegersupermatrixsemivectorspaceofrefinedlabels overZ+ ∪ {0}. a)FindalinearoperatorTonVwhichisnoninvertible. b)WriteVasadirectsumofsemivectorsubspacesof refinedlabelsoverZ+ ∪ {0}.
269
49.LetM= 123 456 789 101112 131415 i 161718 192021 222324 252627 282930 aaa aaa aaa aaa aaa aQ{0} aaa aaa aaa aaa aaa LLL LLL LLL LLL LLL LL;1i30 LLL LLL LLL LLL LLL + ∪ ∈≤≤ bea supermatrixsemivectorspaceofrefinedlabelsoverthe semifield Q{0} L + ∪ . a)FindthedimensionofMoverQ{0} L + ∪ . b)WriteMasapseudodirectsumofsemivector subspacesofrefinedlabelsoverQ{0} L + ∪ . c)WriteM=i i Wasapseudodirectsumofsuper matrixsemivectorsubspacesofrefinedlabelsover Q{0} L + ∪ . d)ShowinMwecanhavemorenumberoflinearly independentelementsthanthecardinalityofthebasisof Mover Q{0} L + ∪ . 50.LetV= 1234 245 3i 759 101 aaaa aaa aaR aaa aa LLLL L0LL 00L0LL;1i11 LL0L 0LL0 ∈≤≤ bea supermatrixvectorspaceoverLR.
270
aaaaaaaa
aaaaaaaa aaaaaaaa
a)FinddimensionofVoverLR. b)Write ∪ Wi=Vasapseudodirectsumofsubspaces. c)IsthenumberofsubspacesofVoverLRfiniteor infinite? d)Givealinearlydependentsubsetoforder10. 51.LetV= 1234 25 i 36 47 aaaa aa aR{0} aa aa LLLL L00L LL;1i7 L0L0 LL00 + ∪ ∈≤≤ be agroupsupermatrixsemivectorspaceofrefinedlabelsover multiplicativegroupG=R L + . a)WhatisthedimensionofVoverG? b)Provethereexistinfinitenumberofpseudosetsuper matrixsemivectorsubspacesofVoverLR. c)Provethereexistsinfinitenumberofpseudosemigroup supermatrixsemivectorsubspacesofVoverLR. d)ProveVisnotasimplegroupsupermatrixsemivector spaceoverLR. e)WriteV= 7 i i1 W = asdirectsumofsubspacesofV. f)WriteV= 6 i i1 W = aspseudodirectsumofsubspacesofV 52.Let V= 1471013161922 i 2581114172023 3691215182124
aR
LLLLLLLL LL; LLLLLLLL 1i24 LLLLLLLL ∈ ≤≤ beasuperrowvectorspaceoverLRofrefinedlabels. a)WhatisthedimensionofV? b)FindR L Hom(V,V). c)FindT:V → VsothatT 2=T. d)FindT 1:V → Vsothat2 1 T=(0).
271 53.LetP= 1111 2222 3333 i 4444 5555 6666 aaaa aaaa aaaa aR{0} aaaa aaaa aaaa LLLL LLLL LLLL ,,,LL;1i6LLLL LLLL LLLL + ∪ ∈≤≤ beasetsupermatrixsemivectorspaceofrefinedlabelsover thesetS=nnn 2{0}3{0}5 LLL ∪∪ ∪∪ . a)FindsubspacesofP. b)WhatisdimensionofPoverS? c)ExpressP= 4 i i1 P = asadirectsumofsubspacesofV. d)DefineT:P → PwithT.T=T. e)DefineT 1:P → Psothat1 1 Texists. 54.Let V= 12 234123 34 234145 56 332246 78 442356 910 aa aaaaaa aa aaaaaa aa aaaaaa aa aaaaaa aa LL 0LLL0LLL LL L0LLL0LL LL ,, LL0LLL0L LL LLL0LLL0 LL } i aR{0} LL;1i6 + ∪ ∈≤≤ beaintegersetsupermatrixsemivector spaceofrefinedlabelsoverthesetofintegersS=3Z+ ∪ {0} ∪ 2Z+.Suppose W= { 12345678910 aaaaaaaaaa LLLLLLLLLL,
272 12 123 34 456i 12 789 34 aa aaa aa aaaaR{0} aa aaa aa LL LLL LL LLL,LL;1i6 LL LLL LL + ∪ ∈≤≤ beanintegersetsupermatrixsemivectorspaceofrefined
55.Let M= 12345 678910 1112131415 1617181920 2122232425 2627282930 3132333435 3637383940 4142434445 4647484950 aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa aaaaa LLLLL LLLLL LLLLL LLLLL LLLLL LLLLL LLLLL LLLLL LLLLL LLLLL i aQ{0} LL; 1i50 + ∪ ∈ ≤≤ bea supermatrixvectorspaceofrefinedlabelsoverQ{0} L + ∪ . a)WriteM=i i W;Wisupermatrixvectorsubspacesof refinedlabelsover Q{0} L + ∪ ofMasadirectsum.
labelsoverthesetS=3Z+ ∪ 2Z+ ∪ {0}. a)FindalineartransformationTfromVintoWsothatT -1 exists. b)FindthealgebraicstructureenjoyedbyHomS(V,W). c)FindT 1andT 2twolineartransformationfromVintoV suchthatT 1.T 2isanidempotentlineartransformation. d)FindtwolineartransformationL1andL2fromWinto WsothatL1.L2=L1.L2butL1.L2=L1.L2 ≠ I.
273 b)LetW= 12345 678910 i 1112131415 1617181920 aaaaa aaaaa aQ{0} aaaaa aaaaa 00000 LLLLL 00000 LLLLL 00000LL; 000001i20 00000 LLLLL LLLLL 00000 + ∪ ∈ ≤≤ ⊆ MbeasupermatrixvectorsubspaceofMofrefined labelsover Q{0} L + ∪ . FindT:M → MsothatT(W) ⊆ W. c)WhatisdimensionofW? d)FindT 1,T 2:M → Msothat(T 1.T 2)(W) ⊆ W. 56.Obtainsomespecialpropertiesenjoyedbysetsupermatrix vectorspaceofordinarylabelsoverasuitableset. 57.LetV= {} 12345i aaaaaaR LLLLLLL;1i5 ∈≤≤ be asuperrowmatrixvectorspaceofrefinedlabelsoverLR. WillV ≅ LR × LR × LR × LR × LR? Justifyyouranswer. 58.Howmanysuperrowmatricesofdimensionsevencanbe builtusingtherowmatrix () 1234567 aaaaaaa LLLLLLLofrefinedlabels? 59.ProveordisprovethatifVisasupermatrixvectorspaceof dimensionnofrefinedlabelsdefinedoverLRthenVis isomorphicwithallsupermatrixvectorspaceofdimension nofrefinedlabelsdefinedoverLR.
274
7
L L LLLLLL LLL LLLLLL L V,LL, LLLLLL LLL LLLLLL L L = } i aR{0} LL;1i24 + ∪ ∈≤≤ and
() 123 1234567
aaaaaaa
LLL ,LLLLLLL, LLL
60.LetV= 123 456 789i 101112 131415 aaa aaa aaaaR aaa aaa LLL LLL LLLLL;1i15 LLL LLL ∈≤≤ beasuper matrixvectorspaceofrefinedlabelsdefinedoverLR.W= 14567 2891011i 312131514 aaaaa aaaaaaR aaaaa LLLLL LLLLLLL;1i15 LLLLL ∈≤≤ bea supermatrixvectorspaceofrefinedlabelsdefinedoverLR. IsW ≅ V? 61.Let 1 2 159131721 312 2610141822 434 3711151923 556 4812162024 6
a a aaaaaa aaa aaaaaa aaa aaaaaa aaa aaaaaa a a
W=
456 aaa
aaa
275 123456 789101112 i 131415161718 192021222324 aaaaaa aaaaaa aR{0} aaaaaa aaaaaa LLLLLL LLLLLL LL;1i24 LLLLLL LLLLLL + ∪ ∈≤≤ beagroupsupermatrixsemivectorspaceofrefinedlabels overthemultiplicativegroupG=R L + . a)IsV ≅ W? b)FinddimensionV. c)FindT:V → WsothatT -1exists. d)FindT 1:V → WsothatT 1.T 1=T 1 e)FindpseudosetsupermatrixsemivectorsubspacesM andPofVandWrespectivelyoverthesetB= nn 11R 25 LLL∪⊆+ suchthatM ≅ P. f)Definef:V → Wsothatf(M)=P. 62.Let 12789123 34101112456 56131415789 aaaaaaaa aaaaaaaa aaaaaaaa LLLLLLLL VLLLLL,LLL, LLLLLLLL = 123 456 7891234 1011125678 i 1314159101112 16171813141516 192021 222324 aaa aaa aaaaaaa aaaaaaa aR{0} aaaaaaa aaaaaaa aaa aaa LLL LLL LLLLLLL LLLLLLL ,LL;1iLLLLLLL LLLLLLL LLL LLL + ∪ ∈≤ 24 ≤ beasemigroupsupermatrixsemivectorspaceofrefined labelsoverthesemigroupR{0} L + ∪ underaddition.
276
d)FindM:V
+ ∪ . e)Findsubsemigroupsupermatrixsemivectorsubspaceof
f)FindpseudosetsupermatrixsemivectorsubspacesofV
L + ∪ . 63.Giveanexampleofagroupsupermatrixsemivectorspace ofrefinedlabelswhichissimple. 64.Obtainsomeinterestingpropertiesaboutgroupsupersquare matrixsemivectorspaceofrefinedlabelsoverthe multiplicativegroupQ L + . 65.LetV= (){ 12345678910 aaaaaaaaaa LLLLLLLLLL } i aR LL;1i10 ∈≤≤ beasuperrowmatrixvectorspaceof refinedlabelsoverLR. a)Findforf:V → LR,alinearfunctionaldefinedby f () 12345678910 aaaaaaaaaa LLLLLLLLLL = 1210 aa...a L; +++ thekerf. b)FindRLR L(V,L). c)WhatisdimensionofRLR L(V,L)? d)WhatisthealgebraicstructureenjoyedbyRLR L(V,L)? e)WhatisthedimensionofVoverLR? f)FindR L Hom(V,V). g)Whatisthealgebraicstructureenjoyedby R L Hom(V,V)?
a)FinddimensionV. b)FindT:V → VsothatT.T=T. c)FindP:V → VsothatP 2=0.
→ VsothatkerMisasemigroupsuper matrixsemivectorsubspaceofVofrefinedlabelsover theadditivesemigroupR{0} L
refinedlabelsofV.
ofrefinedlabelsoversubsetsoftheadditive semigroupR{0}
277 66.FindsomeinterestingpropertiesenjoyedbyRLR L(V,L)= {f:V → LR}whereVisasupermatrixvectorspaceof refinedlabelsdefinedoverLR. 67.FindthealgebraicstructureenjoyedbyHS(V,V)whereVis asetsupermatrixsemivectorspaceofrefinedlabelsdefined overthesetS= R{0} L + ∪ . 68.LetV= 1234 5678 i 9101112 13141516 aaaa aaaa aR aaaa aaaa LLLL LLLL LL;1i16 LLLL LLLL ∈≤≤ bea supermatrixvectorspaceofrefinedlabelsoverLR. W= 234 i 256 357 ii 467 aaa aR aaa aaa aa aaa 0LLL LL; L0LL 1i7; LL0L LL LLL0 ∈ ≤≤ =− beasupermatrixvectorspaceofrefinedlabelsoverLR. a)IsV ≅ W? b)FinddimWoverLR. c)FinddimVoverLR. d)StudyR L Hom(V,W)andR L Hom(W,V). e)FindT:V → WsothatT.T=T. f)FindT 1:V → Wsothat1 1 Texists.
278 69.LetV= 12 34 56 1234 78 5678 910 9101112 1112 13141516 1314 17181920 1516 1718 1920 aa aa aa aaaa aa aaaa aa aaaa aa aaaa aa aaaa aa aa aa LL LL LL LLLL LL LLLL LL LLLL LL,, LLLL LL LLLL LL LL LL 123456 78910111213 14151617181920 aaaaaa aaaaaaa aaaaaaa LLLLL0L LLLLLLL LLLLLLL , 1234 5678 9101112i 13141516 17181920 aaaa aaaa aaaaaR aaaa aaaa 0LLLL L0LLL LL0LLLL;1i20 LLL0L LLLL0 ∈≤≤ beasemigroupsupermatrixsemivectorspaceofrefined labelsover R{0} L + ∪ underaddition. a)FinddimV. b)Find R{0} L Hom(V,V) + ∪ . c)FindW1,…,WtinVsothatV= ∪ WiwhereWi’sare semigroupsupermatrixsemivectorsubspacesofVover R{0} L + ∪ whereWi ∩ Wj= φ ifi ≠ j1 ≤ i,j ≤ t. 70.Findsomeniceapplicationsofsetsupermatrixsemivector spacesdefinedoverthesetS=n2{0}R{0} LL + ∪∪ ⊆ .
279 71.Doesthereexistsasetsupermatrixsemivectorspacewhich
72.Doesthereexistasemigroupsupermatrixsemivectorspace
73.Proveeveryintegersetsupermatrixsemivectorspaceof refinedlabelshasintegersubsetsupermatrixsemivector subspaceofrefinedlabels! 74.LetV= 1111 1111 1 1111 1111 aaaa aaaa aQ aaaa aaaa LLLL LLLL LL LLLL LLLL ∈ beasuper matrixvectorspaceofrefinedlabelsoverthefieldLQ. a)CanVhavenontrivialsupermatrixvectorsubspaceof refinedlabels? b)FindQ L Hom(V,V). c)DoesT:V → VexistsothatT 2=T?(T ≠ I). d)FindQLQ L(V,L). 75.LetV= 11111 11111 11111 111111 11111 11111 11111 aaaaa aaaaa aaaaa aaaaaaR{0} aaaaa aaaaa aaaaa LLLLL LLLLL LLLLL LLLLLLL LLLLL LLLLL LLLLL + ∪ ∈ bea semigroupsupermatrixsemivectorspaceofrefinedlabels overtheadditivesemigroupR{0} L + ∪ .
hasnosubsetsupermatrixsemivectorsubspaceofrefined labels?Justifyyourclaim.
ofrefinedlabelswhichhasnosubsemigroupsupermatrix semivectorsubspaceofrefinedlabels.
280 a)FinddimV.
+ ∪ ? c)CanVhavenontrivialsubsemigroupsubmatrix
+ ∪ ? d)Findpseudosetsupermatrixsemivectorsubspacesof
+ ∪ . e)Find R{0} L
+ ∪ . 76.LetV= 1111111 1111111 1 1111111 1111111 aaaaaaa aaaaaaa aR aaaaaaa aaaaaaa LLLLLLL LLLLLLL LL LLLLLLL LLLLLLL ∈ beasupermatrixvectorspaceofrefinedlabelsoverLR. a)CanVhavepropersupermatrixvectorsubspacesof
? b)IsVsimple? 77.LetV= 1111 1111 1 1111 1111 aaaa aaaa aQ aaaa aaaa LLLL LLLL LL LLLL LLLL ∈ bethesuper matrixvectorspaceofrefinedlabelsoverLQ. a)IsVsimple? b)FindQLQ
78.LetV= 111 1111 111 aaa aaaaQ{0} aaa LLL LLLLL LLL + ∪ ∈
b)CanVhavenontrivialsemigroupsubmatrixsemivector subspacesofrefinedlabelsoverR{0} L
semivectorsubspacesofrefinedlabelsover subsemigroupsinR{0} L
refinedlabelsoversubsetsin R{0} L
Hom(V,V)
refinedlabelsoverLR
Hom(V,L).
supermatrixsemivectorspacesrefinedlabelsoverthe multiplicativegroupR L + . a)IsVsimple?
281
beasupermatrixsemivectorspaceoverthesemifieldQ{0} L + ∪ of refinedlabels. a)FinddimofVoverQ{0} L + ∪ . b)IsVsimple? c)Find Q{0} L Hom(V,V) + ∪ . 79.LetV= 11111 11111 1 11111 11111 aaaaa aaaaa aR{0} aaaaa aaaaa LLLLL LLLLL LL LLLLL LLLLL + ∪ ∈ bea supermatrixsemivectorspaceofrefinedlabelsover R{0} L + ∪ . a)IsVsimple? b)CanVhavesubstructuresifinsteadofR{0} L + ∪ ;Vis definedover R{0} L + ∪ . c)Find R{0} L Hom(V,V) + ∪ . d)FindT:V → VsothatT -1exist.(Isthispossible). e)CanT:V → VbesuchthatT.T=T(T ≠ I)? 80.LetV= 1111 1111 1111 11111 1111 1111 1111 aaaa aaaa aaaa aaaaaR{0} aaaa aaaa aaaa LLLL LLLL LLLL LLLLLL LLLL LLLL LLLL + ∪ ∈ beagroup
282 b)DoesVhavesubgroupsupermatrixsemivector subspaces? c)DoesVcontainpseudosemigroupsupermatrix semivectorsubspaces? d)CanVhavepseudosetsupermatrixsemivector subspacesofrefinedlabels? e)Find R L Hom(V,V) + . 81.Characterizesimplesupermatrixvectorspacesofrefined labelsoverLRorLQ. 82.LetV= 1 2 3 4 512 123456 634 789101112 756 8 9 10 11 a a a a aaa aaaaaa aaa aaaaaa aaa a a a a L L L L LLL LLLLLL L,LL, LLLLLL LLL L L L L } i aR{0} LL;1i12 + ∪ ∈≤≤ beaintegersetsupermatrix semivectorspaceofrefinedlabelsoverZ+ ∪ {0}. a)FindWi’sinVsothatV= ∪ Wi;Wi ∩ Wj= φ ifi ≠ j. b)FindZ{0}Hom(V,V) + ∪ . c)IsVsimple? d)IsVfinitedimensional? e)CanVhavesubsetintegersupermatrixsemivector subspacesofrefinedlabelsoverS ⊆ Z+ ∪ {0}?
283 83.FindthealgebraicstructureenjoyedbyR L Hom(V,V) whereV= ()111111 aaaaaaR LL...LLL;LL ∈ . 84.FindthealgebraicstructureofRLR L(V,L)whereVis giveninproblem83. 85.FindthealgebraicstructureofwhereV= 111 1 111 aaa aQ aaa LLL LL LLL ∈ . 86.Find Q L Hom(V,V) + whereV= () 111111 11 111111111 11 111111 aaaaaa aa aaaaaaaaa aa aaaaaa LLLLLL LL LL,LLL,L,LLLLL LLLLLL } 1 aQ{0} LL + ∪ ∈ groupsupermatrixsemivectorspaceof refinedlabelsoverthemultiplicativegroupQ L + . 87.If Q L + isreplacedbythesemigroupQ{0} L + ∪ underaddition, Vbecomesthesemigroupsupermatrixsemivectorspaceof refinedlabelsover Q{0} L + ∪ ;semigroupunderaddition. Find Q{0} L Hom(V,V) + ∪ . a)Compare Q{0} L Hom(V,V) + ∪ inproblem86with Q{0} L Hom(V,V) + ∪ .
88.LetV=
L L L L L LL L L L L L
1 1 1 1 1 1 1 1 1 1 1
a a a a a aR a a a a a
∈ beasupermatrix(column vector)vectorspaceofrefinedlabelsoverthefieldLR. a)FindR L Hom(V,V). b)RLR L(V,L). 89.Obtainsomeinterestingpropertiesenjoyedby Q L Hom(V,V),Visasuperrowvector,vectorspaceof refinedlabelsoverLQ. 90.Whataretheapplicationsofsupersquaresymmetricmatrix vectorspaceofrefinedlabelsoverLR? 91.Whataretheapplicationsofsuperskewsymmetricmatrix vectorspaceofrefinedlabelsoverLQ? 92.IfVisadiagonalsupermatrixcollectionofrefinedlabels vectorspaceoverLR;canwespeakofeigenvaluesand eigenvectorsofrefinedlabelsoverLR? 93.Canthetheoremofdiagonalizationbeadoptedforsuper squarematrixofvectorspaceofrefinedlabelsoverLR? 94.Studysupermodelofrefinedlabelsusingtherefinedlabels asattributes.
284
285
95.LetV= 123 456 789i 101112 131415 aaa aaa aaaaR aaa aaa LLL LLL LLLLL;1i15 LLL LLL ∈≤≤ beasuper matrixvectorspaceofrefinedlabelsoverLQ. a)IsVfinitedimensional? b)DoesVhaveasubspaceoffinitedimension?Justify. 96.Obtainsomeinterestingresultsrelatedwithhyperspaceof supermatrixvectorspaceofrefinedlabels. 97.Obtainsomeapplicationsofsupermatrixsemivectorspaces ofrefinedlabelsover R{0} L + ∪ or Q{0} L + ∪ . 98.ProveL= 1234 58910 i 6111213 7141516 aaaa aaaa aQ aaaa aaaa LLLL LLLL LLLLLL;1i16 LLLL ∈≤≤ is agroupofsupermatrixofrefinedlabelsunderaddition. 99.ProveasupermatrixvectorspaceofrefinedlabelsoverLQ orLRisneverasupermatrixlinearalgebraofrefinedlabels overLQorLR. 100.LetV= {} 128i aaaaQ LL...LLL;1i8 ∈≤≤ bea superrowmatrixvectorspaceofrefinedlabelsoverLQ. Visofdimension8. Howmanysuperrowmatrixvectorspacesofrefinedlabels ofdimension8canbeconstructedoverLQ?
286
101.IfV={m × nsupermatrixwithentriesfromLQ}beasuper matrixvectorspaceofdimensionmnoverLQ. Howmanysuchspacesbeconstructedwithmnaturalrows andnnaturalcolumns? 102.LetT= 1234 58910 6111213 7141516 17181920 aaaa aaaa aaaa aaaa aaaa LLLL LLLL LLLL LLLL LLLL be5 × 4matrixofrefined labels.InhowmanywayscanTbepartitionedtoform supermatrixofrefinedlabels. 103.Doesthepartitionofamatrixofrefinedlabelsaffectthe basisalgebraicstructure? 104.IfV= 12310 45611i 78912 aaaa aaaaaR aaaa LLLL LLLLLL;1i12 LLLL ∈≤≤ and W= 1234 5678i 7101112 aaaa aaaaaR aaaa LLLL LLLLLL;1i12 LLLL ∈≤≤ betwo supermatrixvectorspaceofrefinedlabelsoverLR.IsV ≅ W? 105.Let V= 12345 678910 i 1112131415 1617181920 aaaaa aaaaa aR aaaaa aaaaa LLLLL LLLLL LL;1i20 LLLLL LLLLL ∈≤≤ and
287
{}
121 34
21 432 aaa
aa aaa
= 123 412 341
1324 41 2123 aaa aaa aaa aR aaaa aa aaaa L0L0L0 0L0L0L L0L0L0 LLLL00LL;1i4 0L00L0 L0LL0L
123410 i 1112131420 aaaaa aR aaaaa LLLL...L WLL;1i20 LLLL...L =∈≤≤ betwosupermatrixvectorspacesofrefinedlabelsoverLR. a)IsV ≅ W? b)IsdimV=dimW? c)IsR L Hom(V,V) ≅ R L Hom(W,W)? 106.LetV= ()
12345i aaaaaaR LLLLLLL;1i5 ∈≤≤ andW= 12345 12345i 12345 aaaaa aaaaaaR aaaaa LLLLL LLLLLLL;1i5 LLLLL ∈≤≤ betwosupermatrixvectorspaceofrefinedlabelsoverLR? a)IsV ≅ W? b)IsRLR L(V,L) ≅ R LR L(W,L)? c)IsR L Hom(V,V) ≅ R L Hom(W,W)? 107.LetV=
i
aa aR
LL0L 0LL0 LL;1i4 L00L 0LLL ∈≤≤ andW
i
∈≤≤ betwosupermatrixvectorspacesofrefinedlabelsoverLR?
a)IsdimV=dimW?
b)IsV ≅ W?
c)FindR L Hom(V,W).
d)IsR L Hom(V,V) ≅ R L Hom(W,W)?
e)IsRLR L(V,L) ≅ R LR L(W,L)?
f)FindabasisforVandabasisforW.
288
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292
B
Abeliangroupofsupercolumnvectorsofrefinedlabels,112-4
D
Diagonalsupermatrix,12-3
Directsumofintegersupermatrixsemivectorsubspacesof refinedlabels,190-7
Directsumofsupersemivectorsubspacesofrefinedlabels, 168-173
Directsumofsupervectorsubspacesofrefinedlabels,143-9
Directunionorsumofsuperrowvectorsubspaces,104-5
DSmfieldofrefinedlabels,34-5
DSmfieldofrefinedlabels,32-3
DSmlinearalgebraofrefinedlabels,34-6
DSmsemifield,53-5
F
FLARL(DSmfieldandLinearAlgebraofRefinedLabels),32-4
G
Groupofsupercolumnvectors,27-8
Groupofsupermatrices,28-30
Groupofsuperrowvectors,25-6
Groupsupermatrixsemivectorspaceofrefinedlabels,236-9
293
Groupsupermatrixsemivectorsubspaceofrefinedlabels,237242
H
Heightofasupermatrix,8-9
I
Integersupercolumnsemivectorspaceofrefinedlabels,181-9
L
Linearfunctionalofrefinedlabels,153-7
Lineartransformationofsetsupermatrixofsemivectorspaces ofrefinedlabels,211-8
Lineartransformationofsupervectorspaces,110-3
Linearlyindependentsupersemivectorsofrefinedlabels,173-8
Linearlydependentset,106-7
Linearlyindependentset,106-7
M
M-matrixofrefinedlabels,96-8
N
Nonequidistantordinarylabels,31-3
O
Ordinarylabels,31-3
P
P-matrixofrefinedlabels,98-100
Pseudodirectsumofintegersupermatrixsemivectorsubspaces ofrefinedlabels,190-8
Pseudodirectsumofsupervectorsubspacesofrefinedlabels, 144-8
294
Pseudosemigroupsupermatrixsemivectorsubspaceofrefined labels,243-8.
Pseudosimplesemigroupsupermatrixsemivectorspaceof refinedlabels,230-5
Pseudosimplesupercolumnvectorsubspacesofrefinedlabels, 125-7
Q
Qualitativelabels,31-3
R
Refinedlabels,31-3
S
Semifieldofrefinedlabels,166-9
Semigroupofsupercolumnvectors,52-4
Semigroupsupermatrixsemivectorspaceofrefinedlabels,220229
Semigroupsupermatrixsemivectorsubspaceofrefinedlabels, 222-9
Semivectorspaceofcolumnsupervectorsofrefinedlabels, 174-9
Semivectorspacesofsupermatricesofrefinedlabels,180-5
Setsupermatricessemivectorspaceofrefinedlabels,200-8
Setsupermatrixsemivectorsubspaceofrefinedlabels,201-8
Simplematrix,9-10
Subgroupsupermatrixsemivectorsubspaceofrefinedlabels, 240-6
Submatrices,8-9
Subsemigroupsupermatrixsemivectorsubspaceofrefined labels,230-5
Subsetsupermatrixsemivectorsubspaceofrefinedlabels, 204-8
Supercolumnvectorsofrefinedlabels,39-42
Supercolumnvectors,9-11
Superhypersubspaceofrefinedlabels,159-163
295
Supermatrix,7-10
Superrowmatrixofrefinedlabels,37-8
Superrowvectorspaceofrefinedlabels,101
Superrowvectorsubspaceofrefinedlabels,103-5
Superrowvectorsofrefinedlabelsofsametype,37-9
Superrowvectors,9-10
Supersemivectorspaceofrefinedlabels,165-8
Supersemivectorsubspacesofrefinedlabels,169-173
Supervectorspaceofm × nmatricesofrefinedlabels,134-6
Supervectorsubspacesofm × nmatricesofrefinedlabels, 140-3
T
Transposeofasupermatrix17-18
U
Upperpartialtriangularsupermatrix,14-15
Z
Z-matrixofrefinedlabels,96-98
296
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