Issuu on Google+

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


พีชคณิต