MA 2 2 6
ihlU 1.1.1 :
2
MA 226
r aI2 . . . aIn 1 aI2 . . . aIn
A, =
+di?ailJ 1.1.3 : Az
MA 226
~a,“?eh
1.1.4
A = [aid
A = Ii -:
A
=
aI,
=
1,
aI2
=
0,
aI3
=
4
azl
=
2,
al2
=
-5,
a2)
=
9
tdkl
1 :)jnh =
i
= 1, L,...,m
j
=
1,2,...,n
(-:
; :)
MA 226
iYaiaof.ha
1.1.6 : 'It? A =
Tr(A)
=
1+5+9 = 15
5
x2
=
2
y2
=
0
1.2.2 nmmudnr’
(Addition of Matrices)
u’u1u
1.2.2 : 81 A = [ad URt B = [bij]
A LL53X B 9XLM’7nyu
~&wflumuo4raminml
A+B = C =
&I.LA+B
Wh<l
[Cij]
c
= c,,
Id Cij = aij+ bij ~50 i = 1,2 ,..., m URZ j = 1,2 ,..., n
=
= a,,+b,,
=
3+1
=
4
Cl2 =
a12+b,, =
3+0 = 3
Cl3
=
a13+b,, =
(-.l)+(-2) =
c21
= a2,+b2,
=
7+3
- 3
= 10
MA 226
n’?Oh 1.2.3 :
=
=
a22
+b22
c23
=
=23+h
000
i 1 0
02.3
Cl2
0
0’
=
03.2
=
3+1
=
3
=
3+4
=
7
000 [ 1 00
00
00
lIU1WlIQJ A ttfWh~iiA+O = ‘A t5l’iIctiuntuvl~n~
0 hiju
identity ilnhllmJ??l
tw~lttijutUQlj:n~~dtilr~Yann’ytuoljn~’dutthru’rii18u~i.n~~u 61 A+ B = 0 t&a L3l5t~ndla’htUvl~n~ tuafn4 A ttficocl~ikpTnw&
A+B
B tihht?oia (inverse) Fh?h77UaTllIoJ
-.4 tu’uuttnutuol3na’ =
3 td&uhl tti3r
0
&&
B bi72~
A+(-A)
=
0
&Oh 1.2.4 :
1%
A = [ -: : -:I B
=
-A
El
&&~A+B
MA 226
= - 12
= A+(-A)
= 0
-4 -3
-32 1
(Subtraction of Matrices)
wâ&#x20AC;?UlU 1.2.4 :
A-B
A-B
8
=
=
A+(-B)
AI--B) =
MA 226
ifad 1.2.6 : t$ B =
38
7Lxhln
=
(-l)A = -A
1.2.5 "lJ~m~Lw~19601n~n~ (The
~t3U.l
Multiplication of Matrices)
1.2.6 : t?'-;
? = [a,j] 6hlWt~Tlflf~Wl
llul@l n xp wk3~KIAuoWJpl~n~
mxn LLAZ B = (b,i] lif)~UJVl~fVa
A LLR: B &I AB = C = [ci,l hk
c,, =. i a- bk, = a,,b,,+aizb2,+...+a,,b,j t=, Ik id8 i = 1,2,. .,m Lb% j = I,2 ,..., p
MA 226
9
aIt
a,,
. . . aIn
a2]
a22
. . a2”
b II
btz
r
s , . b,j
... b tp -
Cl1
Cl2
. . . c1p
... b2p
C2I
c22
. . .
czp
= ait
amI
ai
am2
. . . ai,
cij cl.
... b np
. . . amn
‘J:L~U~-l21U-lFIVOJ
A 60 2 x 3 LtRfVUlWdOJ
CII =
=
[Cij]
&aIkbkl =
=
Cli 1 1Cl2 Cl1
Gil2
6OURQTttWNL%JV6ni
10
=
A
IU;tu’i = 1.2 LWj = 1,2
c22
a,Ib,,+a,2b2,+a,lbsl =
(2)(3)+(-1)(l)+(3)(2) =
~12 = ,$,atkbk2 = attbt2+at2b22+a,3b32 = (2) (0)+(-l) (4)+(3)(2) 3 k;,a2kbkl
. . ’ cmp
B ?%I 3 x 2 &hWMQnt AB Ml\Zi
tY+~ltilU?UWl6Jud%DY A = iW?Uttn?lIEN B = 3 ii?‘LziC LA UR: B UUclO AB = C 6&UlWOJ C 6% 2 x 2 UR: c
Glt
a2,btt +az2bzt
+a2h =
C21
=
~2
. = ,~,adk = a2tbt2+a22b22+a23b32
=
II
=
2
(4) (3)+(2) U)+(O) (2) =
14
(4) (0) +a (4)+(o) (2) = 8
MA 226
6Jitl.i A B
=
2
-1
4
3
2 0
'lun'luoJlQo7n'uo=riiu~:l~
.[ 1 3 0
B A =
1 4
2 2
2
4
eYai?bh 1.2.8 :
MA 226
11
ocrAui-I
12
AB = 0 h&i
A
#
0 LLR::
B f 0
MA 226
iih.il~
1.2.10 :
1s
&IL La uuao
MA 226
A
=
L
1
0
- 2
3
1
1
1
aij aji
tiluruo13nB4Jijvul”
A’
=
[i]
5l~iJllU7RtilU 3 X2 LLRt
a:,
=
alI
=
1,
a;,
=
aI2
=
a:*
=
azl
=
3,
a:,
=
a22
=
A’
=
2 x 3
=
0,
a\,
=
al3
=
- 2
1,
a\,
=
aI3
=
1
1 3
2.
14
I< A = [: : :],
B = [; ;I, C=
(1) C + E
(2) 2C - 3E
(3) AB UA: BA
(4) CB+D
(5) 3(2A) UN: 6A
(6) AB+D2 UA: D2 = DD
(7) A(BD) UA: (AB)D
(8) A(C + E) LLR: AC + AE (9) 3A + 2A LW SA
(IO) A’
(11)
(13) B’A’
(14) (C+E)’
(A’)’
(12) WV’ (IS) C’+E’
MA 226
(1) (AfB? # A2+2AB+B2 (2) (A+B)(A-B)
j, AZ-B2
10. tb c tih.mnairl~ 9 tttx A, B tilutu~jnd?‘q~~kuulntall?n’u (1)
Tr(cA)
=
cTr(A)
(2)
Tr(A + B)
=
Tr(A) + Tr(B)
(3)
Tr(AB)
=
Tr(BA)
MA 226
w~pii~
15
1.3 a~sru~uoJfll~nr=n’l~~~~l~6~~~~~
Algebraic Properties of Matrix Operations
&eh 1.3.1:
II12 -2 1
431 j + I:
-:
-:1
= [:
:
:I = i:
Yltplij 1.3.2 : sl”l A, B LLA::C r%4tlJQljn6W4ln
A+(B+C)
=
-1
-:j + [-:
:
:]
m x n L&J
(AtB)+C
c
f&JOoa’:
I w ” A = /a,,],
Gi”Jh4
1 6
B = /bi,j blRt
A f (B + Cl
C = [q,]
=
[a,, + (b,, + c,,)]
=
[h, + b,,) + c,]
=
(A-tR)+C
MA 226
hh 1 2 I 3 41
1.3.2 +
12 (F 34 I
2
(F 3 +
1 _. 2 1
2
1
I3
__ 2
+
3
2
L-2
I> [ :I = [Ii I+[-: +
=
1I
64 4 3
3
1
-2
1I %
=
&&.t
1 2 i 3 41
MA 226
+
2 (1-3
i I i 1 64 3 4
1 - 2 11
17
ciative Law for Multiplication)
nqnij 1.3.4:
IK r LLt% s tilu~tnm%I
tilutu~fnQuuw
r(sA)
=
= [aid LLAZ B
s(rA)
= [bij] [r(Saij)]
=
[da,,)]
=
[s(raij)]
=
s(rA)
rB
=
[rbij]
A(rB)
=
(2)
=
=
(rs)A
r(AB)
1.3.3 :
b?
A
=
WA)
=
L 1
4
2
2l’:
=
18
=
=
r(sA)
(1)
thidl~
(rs)A
A(rB) = r(AB)
(2)
A
m x n LLA: B
n x p 1:IG
(1)
GQQd:‘IK
@J IfiA tihJol~n~lIulcl
2
-3
3
4
LLW:
B
=
_; ,I] = [:: _ ;; :I] = 6[; -; :] (2x3)A
HA 2 2 6
A(2B)
=
2(AB)
=
&&t
A(2B)
=
2(AB)
mqtdi 13.6 : (1) ni A tttc B ri%uuaJna’ww
mxn ttix C tilutumfnGuu?n
n x p tth ( A + B ) C = A C + B C (2) 0’7 C thtMI~VhU1~ mxn LLAZ A, B thtUR%lhl~
n x p U&l C(.A+B) = C A + C B
I<
A = [aik],
6&t
B = [bi,] A +
(A+B)C
B
UAZ
C = [Ckj]
= [a,+ bi*]
=
[ ,t,(as. + bir)ckj]
= AC+BC
MA
226
19
c;tiedis 1.3.4 : 0
IG;
A = [I
-:
:],
B = [I
; -:]
ttA:: C = [;
2
&&t
-1 I
(A+B)C
=
1 AC-FBC
=
o-
2
2
3
- l -
= i’: J+ [: -:] &lk4
( A + B)C =
+
[ =
0 0
1
2 3
-1
1
o-
2
2
I[ 3
-I-
-18 0
i I 12 3
AC-k BC
omnt-p~ 1.3.5 r5india-h
ni~uanuA::n,la~nrtua~n~‘rilurlJ~,lungnls
nX?lU (Distributive Law)
nqrr; 1.3.8 : fil r, s Shbw~naGI~ “1 uf+r A, B Jhkhhm m X” t&-J
GpOd
(1)
(r+s)A
=
rA+sA
(2)
r(A+B)
=
rA+rB
(1) 1% A = [aij]
&l&4
(r+ s)A = [(r + s)aij]
(2) 1%
A
=
[raij + sad
=
rA+sA
=
[aij]
LLWE
B
=
[bij]
key
r(A+ B) = r[aij+ bij]
20
MA 226
= [r(a,j + bij)] = =
[raij + rby] [raij]
=
+ [rbi,]
rA+rB
?lqMjj 1 . 5 7 : $1 A ’ LLR: B ’ &I ~~1~~hWll~JLaJ’hl~ 6rtxnS1 c iionmai3sl
MA 226
A LLR: B 81lUh+h
7 &&A
(1)
(A’)’
=
.4
(2)
(A+B$
=
A’tB’
(3)
(cA)t
=
CA’
(4)
(AB)’
=
B’A’
21
&
(AB)'
=
[1 I1
13 6
A' = [I
B'A'
22
7
;] UR:: B' = [: ", :]
= [; ; ;]
[I
;]
= [;:
;]
= W)'
MA 226
1. P4llt%Y-h A ( B C ) = ( A B ) C lib
A = [: -:I, B = [ -: -: :] tt8: c = 2 .
WllafNil A(rB) = A
=
[;
3. WllWWh A
4.
=
-;],
r(AB) lb
B
=
[-;
-;
-:]
11%
r
=
-3
C(A+ B) = CA+CB ld0
[;
1:
21,
B
=
[;
:
-;j
LlR:
ilJll1619’h (r + s)A = rA+sA lb r = 4, s =
C
=
- 2 LLRE
[_:
A =
5 . il~llffFlJ~l r(A+ B ) = rA+ rB l;O
A
6 .
=
[;
-I],
B
?JllWlJil (ATB)’ = A
[
-!
i]
1lR:
A’kB’ LLR: (CA)’ =
r
=
i 1 2
-3
4
2
-3
CA’ l&
=
7. ~JllR@lJil
A
=
-11
LlAZ
c
=
-4
(AB)’ = B’A’ lib
=
ttat
8
=
8. PJMllWl%la’ A 11% B iJhl41" 9. WllaJ~h~ A # 0 ~JihJUl61
3
-1-
2
4
1
2-
2 x 2 LlR:: A # B # 0 11APh~~ A B = 0
2x 2 ll&l~~ A* = 0
10. W4?18J&l~ A, B 11% C &hl" 2 x 2 LlR~ril~ii
AB = AC h”i B # C
11% A # 0
MA 226
‘23
&llU 1.4.1 : tYl~:t%JfN%JtolffPl?
A 'h dU symmetric 61 A = A’
(sv?tih 1.4.1:
A
=
tih symmetric matrix
A
LW(flrh
I
=
A’
%llU 1.4.2: t3TJ~t~~llt%Jfl~fl~ A iI skew-symmetric 61 A = -A’
i?d"aeh
1.4.2:
o-4 4
I<
A
=
0
o-1
i- 3
24
0
l-2 0
2-5
3
5 0
MA 226
-3 A’
rz
1
2
0
-5
5
0 I
0
1
-2
0
5
-5
0
= 0
nqnij 1.4.1 :
!4JAU?flllOJ
ii? A
A
3I
t9u6rJ~Sna’~q~~l~ 7 Lm1ulx-lAuu
A 1JiotjlupJ
symmetric matrix LtR: skew-symmetric matrix 16 ttR~P3fh.4MtGlJ
tt~?Jt~U?LV’h~U
i%.&O A = S+ K T’Nd S <O symmetric matrix LLfC K %I
skew-symmetric matrix
fiql$ ,S = :(A+ A’) ttt% K = :(A - A’) 'W%4h S th symmetric LLt% K tau skew-symmetric &<LW~lZ’h s'
LtA:
-K’
=
= -[ zz
MA 2 2 6
[!-(*+A*)]'
$+A’)]’
=
;(~+A31
=
$A'+A)
=
s
= - ;(A-& = - ;(A’-&
$A-A’) = K
25
th skew-symmetric &tdO~d tt.l.Wt&j?
LLAE
W&lJw^~?6’h A L&U~t%l~?UJd
hlWJJukh A Rlul~ntQUU’l~~~~UJ~~~~~nYOJ
symmetric matrix
A = S1 + KI I@lUd S, L&i symmetric
skew-symmetric matrix 1% flttlJLlMdJb%
LLA: K &I skew-symmetric
S + K %fLw’UJ
LflTiZdhJ~$‘h
S+K = S, +K, U’UtO %&lJ
fiQ'lii1 S = S, URE K = K, ‘3111
A
=
S,+K,
+YJ&
AL
=
(S,+K,)'
=
S,-K,
=
S;+K;
t?‘dTlE’h
S1 till.4 symmetric <JI.?U S: = S, LLAt: K, th skew-symmetric 6JlfU K: = -K, tts3: A+A’ = (S,+K,) + (f&-K,) = 2S,
s, A-A'
tmr
=
= (S,+K,)
K, =
;;Ju”U
~(A+A')
= s
- (S,-K,) = 2K,
~(A-A')
=
K
PY-meilS 1.4.3:
14,
s
2 6
=
tmt
K =
i)fthh
A
=
+(A+A')
=
1
712
3/2
712
6
3/2
312
3/2
3
S+K
MA 226
GQlU 1.4.4:
61
A = [aij]bb&conjugate 9109 A GO A = [ail] dQ aIj ;O conju-
gate 9104 aij
cYaix.i~s
1.4.4 :
M
A
= [,ii
:,:,I
&ispdu
A = [ z_ii 13::] iTat& 1 . 4 . 5 :
[ 1 12
Iii
MA 226
A : =
1
03-1 2 1 -3
27
t!tLhrjl
fiUl~ll~nor?lloJ
Vh?u”ULOJ LdU 2 = 2+0 ,i
03iU<l
A lihilU?W?JLLRt: conjugate ?l0Jh?U~fJ?%Q conjugate 110J 2 &I 2 = 2 - 0 i = 2 ‘&lfU A = A
?lqMa ,1.4.2: A Lh real matrix 7%0&l A = A
&a&s 1.4.13 :
M A
=
[2:,i
ihlal 1.4.5 : ~TKCzL%JfW~~rl~ A
&dlJ 1.4.7 : A =
28
z
:iZIJ
h&4
Hermitian
61
A = A*
Lfh Hermitian matrix
MA 226
ihlar
hWil4
1.4.6:
A h!h idempotent 81 A2 = A
t~l5I:l~~fllXJol~fl~
1.4.8:
fil
A =
A*
=
[:
“oi[:
;]
=
[;
“bl
=
A
$.$I4 A b%4 idempotent matrix
&Jlal
1.4.7:
1,51'%L%fW@l~fla
~lUX.4L&.JY?fl’~~
‘j Ut3SItL%Jn
A h&4 nilpotent 61 A” = 0 L&l k tifll.4 A 41 nilpotent of index k t% A’ = 0 tt@!
A’-’ # 0
tih.h 1.4.9: $1
A
=
A* =
[-: -:][-: -:] =
[I ;] I=OLLREA#O
gaI?U A L%-4 nilpotent of index 2
$.llU 1.4.8 :
MA
226
ifI
A I,hLoJ~~n~~~~~
LTliltLifJll
A 'hh diagonal matrix
2 9
9Yadl~ 1.4.10 :
D,
=
?& D, LLFiE D, lh diagonal matrix t~lil~t%lU~Lhh 7 ~lhlOJ diagonal matrix &ii
ihl 1.4.9 :
n'7 A Lh diagonal matrix L7lll:L~Un
A hhRLflRl~L%J~fd
(scalar matrix) hiatnal~~aduwuu?t~unuuJyuuoJ A LYilnâ&#x20AC;&#x2122;U
1.4.11 :
iYaiaah
s
=
2
0
0
L0
2
01
0
0
2
SC
30
tih.tatnfiliruvSni
L. 1 C
0 . . . 0
0 :
c 1.. 0
0
o...c
L&I
c
riluatnalrl~ 3
MA 226
Sijflh 1.4.12 :
I
hoi 1.4.10 : Identity matrix #“SO unit matrix ?%%3tUfldUJ~%&J?J
c = 1
I
I” =
hH.il-a 1.4.1s :
MA 226
31
= AI, = A
ihol 1.4.11: tmctâ&#x20AC;&#x2122;iunumfni A 'il involulo:y
51 AZ = I
Lillu upper triangular matrix
32
MA 226
B =
1[ 0 01 -1 1
MA 226
3
0
0
t&4 lower triangular matrix
-5
33
hJol~n~
1. F)J~@dl 2.
A th symmetric U&9 A’ Odh symmetric dhU
ni A tilutu~fn&@%
5GpZ7
(1) AA’ LW A’A t&.4 symmetric (2) A + A’ LOU symmetric (3) A-A’ th skew - symmetric 3. IhJoli’n~ A LW B th symmetric WGigT~tiil (1) A + B t6U symmetric (2) AB l&t symmetric l%ltdO AB = BA 4. P~t!hMJol~n~
A \%$UJLhWlJXlllO~
symmetric matrix LLR: skew-symmetric
matrix t&I 3-2 5
A = I
-1
1 2
3
6
2
I
(1)
(CA)* = EA*
(2)
A = x
(3)
(A)’
(4)
(A*)* = A
(6)
iiii
=
A’
( 5 ) (A+B)* = A * + B *
=
iiE
( 7 ) (AB)’ = B*A*
6. #I A t%.WJ~‘rfdhl
7 %$$a7 A*A LLA: AA* th Hermitian
7. fhJol?n~ H dU Hermitian PJ$fil~~l
B*HB Il:th Hermitian &Xl
8. ~~~~~d~lR~l~n~~~‘lUtt~~t~~~tt~J~~~oJ 9 . 51 A tilUtuAna’Cq?~~J
A = -A* (iiU&~
Hermitian matrix thIh?UO?J aij = -3 t~icxiuntu~3nB
A +h
skew Hermitian PJw^~d+h (1) iA ti)U skew-Hermitian $1 A th Hermitian (2) iA th Hermitian 51 A tfh4 skew-Hermitian
34
MA
226
IO. $1 H th Hermitian LLA:: H = A+iB t&l A LLRt B t&.4 real matrix OJfi@'h A th symmetric LLRZ B th skew-symmetric 11. ~J~ltJJ~~Vh4l@l
2 x2 ttAtti)U idempotent $Ja 1 O~h@lJh~lLI
12. %lJMl diagonal matrix $JhUlW 3 x 3 ttA:Lh idempotent 13. F)JMlLUBI~&‘LlUl@
2x2 LLRtdU involutory
14. Qln (A- I)(A + I) = 0 +dPIHiQbi-h
15. ilJLLRClJ~~t&.&~l
A = I V&l A = -I ilJWlV=hQrilJl.h~fWl
A + B LLRZ AB t&.4 upper triangular 61
S” =
MA 226
35
,A2
alI
aI3 ! aI
[h
i]
aIs
azl a22 a23 i az4 az5 --__--__-__-______ f -____-___-_
A =
a3]
A
36
aI2
=
a32
a33 (1 a34
a3
=
MA 226
A,1 f&l
51
MA
226
A12 A22
Al3 A23 1
B =
3 7
I$‘=
LLRt: B =
A + B
=
------A- - _
. _.-_.- - +.-_-. A,, + ‘31, A21
38
+
B2,
&2+ A22
+
4, B22
1
MA 2 2 6
&& AB =
A,,Jh, +A,,&,
AIIBII + A&,,
.&,BII +Adh
&nBn + A&2
1
9Yah 1.6.7 :
A =
MA
226
LtRt B =
39
h+[:l i: :I+[: :I Liz :I+[: :I1 AB
=
=
40
[o] + [o] [21 + [o]
[o 01 + [o 01 [o o] + [o o]
4:o oi2 0 4 I 610 OiO 0 6 --A---;---- = 010 o:o 0 --* - - - - + --_2 : o 0;o 0 _02 : -
lo 01 + [o 01 10 01 + [o 01
0
0
2
0
0
0
0
0
00
00
00
00 I
a
MA 2 2 6
AB =
7 3 - 4 5
26
2. WHlUR~fU ABOi A = LLR: B =
do
12 =
I 1 1 0
0 1.
31
[ 0 A,1 1 I2
0
B,
0
B,
0
I,
B3
, A , = [3],B,
A =
MA 226
1
=
alI aI2 aI3 [1 azL
a22
a23
a31
a32
a33
41
r 01 01 oiolo0 00;f : ’1 0 0
1 ; 0 0; 1 ---------.-.-------L--------------~---o! 1 0; 0 0 0 0
o!o
1
1
1:o
--_--__-__-_--_-__-
4 2
I i 0 f _-_--_--_-_-L_ I___-
0
0 :,
1
MA 226
GJlol
1.6.1 :
‘lpll’ A
t%.uu6in&iul~ n x n
AB = EA = I, ttt% taiP~ndl?-h ttw&iai?u
A-’ &&I
L511:fdl12h
B
ibitWl%l~ E
B th~Ut+li~IiOJtWl%l~
$J?'hl$:
A tt%33%lU
= A-’ GiJh AA-’ = A-IA = I, n”7tuhviT A Ght-daia
A tfh nonsingular ttd?% A r~J~ettaaiat~lc=ndla~~
A dU
singular
A = [: :I B = [ -: -:I ‘QJ=tV?Wil AB = BA = 1 GJ$U A th^oUt3~i~Y~J
A
A
=
b.bnirKiutaoinuo4 A& t31auuGi~1
MA 226
B LLR: E th~Ut?&WlJ
E
a
b
c
d
=
43
nio
i
a+2c 2a+4c
2;4;]
a+2c 2a + 4c
=
=
[;
;]
1
= 0
&iJuu
b+2d UR:.
=
0
2b f 4d = 1
I
I
w4gp6 :
UUlJ6ibbJ@l%l~ BLLWZ CLih?hL?oiRuo~ A&u AB = BA = I LLR'. tdu cs”3Qdls
B = BI = B(AC) = (BA)C = IC = C
1.8.3 :
A
[ 1
I$ A-’ = a b c
Ot&iOJMl
=
a,b,c, d8&11fl
d
AA-’
4 4
AC = CA = I
=
A“A
=
1
MA 226
....... (1.6.1)
....... (1.6.2) F1lt-1 (1.6.1) FE\&
a+2c 3a+4c
= =
1 0
LLAE
u~~un?~~~~asst~~~~~~~
b+2d
=
0
3b+4d
=
1
a = - 2 , b = 1 , c = 3/2 LEG d = -l/2
q1f-j (1.6.2) ??$I:% a = -2, b = 1, c = 3/2 LLA: d = - l/2 LliU&J?n’u
*-’ = [ ,:
i2]
iTat& 1.6.4 :
87 A LLAt B Vil\rn”lih nonsingular 11&a AB WilU nonsingular
?l$M~ 1.6.2 : LLA:: (AB)-’
MA 226
=
B-IA-’
45
iiigo;
(A@
(B-IA-‘)
(B-IA-‘) 6Jlki u”UiiO
(AB)
=
A(j@ =
,&IA-’
=
A,&-’
=
1
=
B-‘(A-‘A)B
B-‘IB
=
B-‘B
=
I
=
(AB) ( B - I A - ‘ ) = (B-IA-‘) ( A B ) = I B-IA-’ tih?ht?E6tW1J A B
A
=
[; ;]
t
B
=
%&I (AB)-’ =
[: ;]
;
AB
B-IA-’
=
[,;
,:]
A-’ = [ -;,, ,y3] , B-* = [ -;;I -:;;I
I B-LA-' = [-:;: -:;:][w:,3 (AB)-’
46.
=
- 13124
[
10124
5124
-2124
,e,]
= [-:2 -:y = CAB)-’
MA 226
(2)
(3) 2 .
0r_ 01 011 011 1
wJ4plil (*-‘;-I =
(4)
[01 01 00 1 (5) a
b
1 0 a 1
1
A
3. WJi+l (A’)-’ = (A-‘)’
7.
w4wm5nG 2 tuvhd ~JiillU7@ 2x 2 LtAdlU singular tt6iwauantih nonsingular
8.
~JnltSJ@l~?la’2
MA 226
LU41?&!JhlWWi
2x2 t6ii’:th nonsingular tt@kXJ7dU singular
47
4 8
MA 226
Kxlfh~ 1 . 7 . 1 :
18
61
e
&lh
ii7IK
e(A)
e,
=
3RI + R1
=
1
-3
t
0
1
1
0
-8
9
i%nisnr%muttn~ (91lumGi)
tmud
1
ttld
2
e2 &m75nxiimuttna
(mufwcju6)
e3 Gm-mrciimuttn~
(ollumCu6) ttu.4 3
LA uumi
el
=
Ri++Rj
e,
=
cRi
(Cci)
e3
=
CRi + R,
(CCi + c,,
e;’
=
Ri ++ Rj
Cci ++ Cj)
e;’
=
I/cRi
(1 /CCi)
ej’
=
-cRi+Rj
(- ccj + Cj)
(Ci ++ Cj)
b%&
9:tGiAjl I-,, ez, e3 t8unl5n~cn’itt~ytiu~~~
%llU 1.7.2 : $1 A LLA: B t?hW.lol?n~li~l~
mx n t~l?:ndlT’h A row (column)
equivalent Tiil B 61 B b%lnnl5n~chlla.utl~ datiodn’uiclfdfil
MA 226
e;‘, e;’ LLR: e;’ 61lUhii
(011Wl0&~1.$
llUtu96fPii
A
Pi,atiseilS
1.7.2 :
b%aiaefhs
1.7.3 :
'317lklU 1.7.3 'ilLLh4<1
$1 A row equivalent M% column equivalent nâ&#x20AC;&#x2122;y B tttk
A 0: equivalent ??IJ B
HA 226
50
,
(1) A row (column) equivalent n’u A (2)
61 A row (column) equivalent 6J B l&J B ?Z row (column) equivalent kl A
(3)
61 A row (column) equivalent %I B LLR: B row (column) equivalent %I C l&l
A Pt row (column) equivalent 6.l C 2 . PJi?pz~l (1) A equivalent l%J A
3.
(2)
<I A equivalent T%J B l&J B IIT. equivalent <Ll A
(3)
51 A equivalent fk B 1lRE B equivalent l% C l&I A ‘Jr equivalent n’u C
F)JllR@lJ~l
A
4.
A 1lRZ B equivalent f6.4 <‘l
=[:I j;]llRE B
smxtiwuru~^an4
A wnxn’lRuol3nQ B ni
(‘)A=[;-;;-j:
(2)A
MA 226
=
=[; -;;;I
[“‘I,
B
B=[,;;;:
=[‘;;I
51
1.8 nJn?na;Iasa#u
Elementary Matrix
PhdN 1.8.1:
E,
=
[:â&#x20AC;&#x2DC;3
E2
=
[;
-%;y
E,
=
[;;I
E4
=
[i:]
=
[1 1 0 3
ES
0 2 0 0 0 1
52
MA 226
MA 2 2 6
53
nqu~ 1.8.2 : n”7 A LLFiZ B t%WJo1~flh&l~
m x n tth A 0: row equivalent
(column equivalent) n’u B 6hdD B = EkEk_,...EIE,A (B = AE,E,...E,..,E,) t& E,, E2, . . . . E, t%ttU@jn(a’t~Q~6U
&pi:
&lkt
54
!$e LhTllarlatYh~n
9
45 e ( I ) = E
81 e-’ t~U~wtaQ~a~Qsnlana~~l
e ttt%
hti4
EE-’
=
e(E-‘)
=
e(e-:
LLA=.
E-‘E
=
e-‘(E)
=
em’(e(I))
&lb4
EE-’
=
E-‘E
=
e-*(I) = Em’ (I)) =
I I
1
i&i0
E-l
G-htaoimtos E ttart&wain e ttn::
tm:
E-‘’
t~Pttuajna’tdoj~~tt~~t~~~~~
E
=
e-'
t8~ni5nat~ittzlutiio?n’u
MA 226
MA 226
55
echelon
0 0
56
0 0
2
0
0
--I
0
0
0
0
0
0
1
0
0
OQO
I
MA 226
vYa”aedis
1.93 :
[I 1 0 0 0
0 1 0 0 , 0 0 1 0
A
MA 226
=
0 0 0 1
1 1 12001
B
=
.
0 0 1 2 3 0 0 0 0 0
5 7
- 1 0 0 0 - 2
1
0100 c
=
0001
I
4 -
8 -2
0 0 0 0 ' 0
0
0000
0
0
+
4
VlqXI~ 1.9.1 : tm%duuw
m xn IFI 7 9:: row (column) equivalent nâ&#x20AC;&#x2122;YtumSni
row (column) echelon LBUO
58
MA 226
w^9lxultuolln~
&di 3 ttmoon
&A4
c = [cij] t& c ihm3nBdoouo4
c ihtlar~u ( m - l)xn
n’&ttd$U4
A hvhtna
1 ou&#i46 3 n’Yrmfn4 C
row echelon 44 row equivalent %I A
ta.m~nim.m m x n 161
9 ‘Jz row (column) equivalent rihm%vii
row (column) reduced echelon MU0
RI ++
R2
0 0 0 2
1
=rhwQn~o~~uiittwPiJ
0
2
llmu 1
1
0
1
(1,2)
-l/2 1 R, + -R, 2
MA 226
0
0
0
1
0
0
2
1
1 2
-
59
R, + (- 2/3)R,
1 Rj + -R3 2
0
0
0
2
1
0
1
3/2
1
-l/2
! 00
1
0
-*5/3 I = H &Of@@
00
0
1
l/2
r o w echelon
4 row equivalent 6l.l A
R, + (-3/2)R,+R,
dJO$WJd R, .- R,- R,
6 0
row reduce
echelon
MA 226
?lqEl~ 1.9.3 : ?l A &bW~~tl&l~l~
mxn I@ 7 U&?J A 8: equivalent n’y
w&Gb$u~lJ
i
Ik
Ok.@-k)
o(m-k)xk
%n-kb(n-k)
1
$sexiunii normal form
&xlha 1.9.5 :
A
zz -1
1 0
1 2 - l
1 0
21
0
-4
0 1
-2
reduced echelon &i
R, + R,-R, Rj + R,iR, R4 +- R,-R,
R4 +- R,-R,
1 0
1
0 0
l - l
1
3
-3
0
0
0 0 0 0
HA 226
-3 0
2
-1’
61
R, + (- 1/3)R,
0
0
1
-
I
1
0
0
1
-
l
1
000
0
R, + R,-R,
0
0
1
0
-
l
1
000
0
0
000
0
0
U 2x3 0 2*3
6 2
1
0
0
0
c,- (-3)C,+C,
0
1
0
-1
1
c, + 2c,+c,
0
0
0
0
0
0
0
0
0
0
c, + c,+c,
0
1
0
0
0
cs + q-c,
0
0
0
0
0
0
0
0
0
0
1
0
MA 226
MA 226
63
0 0 - l
2 3
02
3
4 5
03
2
4 1 row echelon LLAZ row equivalent I% A
(2) 9WWJW?lI~ C ~xI$I.&~~ row reduced echelon LLAâ&#x20AC;&#x2122;. row equivalent ?%I A
rl 0 17
2.
(1)
64
wmsSnG B 45O$~~lJ column echelon LLRz column equivalent 7%~ A
MA 226