[PDF] Numerical methods of matrix inversion

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[PDF] Numerical methods of matrix inversion - CORE

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NUMERICALMETHODSOFMATRIXINVERSION

KEITHDONALDRUSH

B.S.,KansasStateUniversity,1962

AMASTER'SREPORT

submittedinpartialfulfillmentofthe requirementsforthedegree

MASTEROFSCIENCE

DepartmentofMathematics

KANSASSTATEUNIVERSITY

Manhattan,Kansas

1963

Approvo^dby:

Majoi:'Professor

COREMetadata, citation and similar papers at core.ac.ukProvided by K-State Research Exchange (INTRODUCTION1

INVERSIONOFAMATRIXBYPARTITIONING..

'22

TheInversionofaMatrixbyBordering28

ITERATIVEMETHODSOFMATRKINVERSION36

CONCLUSIONU6

ACKNOVTLEDGEIffiNTIi7

REFERENCES.ii8

INTRODUCTION

methodsofcalculatingtheinverseofamatrix. willbedenotedby

A-(a^g).

ascalarcandamatrixAisgivenby.^ cA"(caj.g). s,thenAB,Thesumoftwomatricesisgivenby

A+B-(a^s+bj.s).

A.B-AB-(21arkbks)k=l

that

AA-1»A"^A=I.

Thedeterminantofa2x2matrixisdefinedtobe

lAl-anai2 agia^2(^11^2"^12^1^' ors(butnotboth)isanoddinteger. ofAcanbedefinedas nn lAl-^a^ic^-21a^s"Jsi»lo"l cedingdefinitionscanbefound. bewritteninthematrixform .AX-G, -1 'bothsidesoftheequationbyA

IX-A'-'-G

X-A'-'-G.

A 1-1-1 2k3 -il-23 21k
elementarycolumnoperations. operations*

PAQ=I,then

a"-'-=QP.-

QPAQ=Q.

QPA=I.

Thereforebydefinitionofmatrixinverse

a""""-QP. forcalculatingtheinverse. B=AI IZ hasbeenreducedtoI B^> IP QZ andPandQhavebeendetermined. onehasobtained B Vm AI <->\n" [zz\MZ thentheworkiscorrectif NMh' duction.

Firstfom

B AI IZ

1-1-1-1I1

I -2h3OlOlOO i -U-23 I1

21UJOOOI

1' I 1 I Iz 1 I 1I

Nowusingelementaryoperations

1 21-2
-U-23 326

C2+ICl

C3+ICl1111

CU+ICl

R2+2R11

Rli-2R11

1 101

121011-2

I1-2-23

-2 1 1 V 1 12 1 6

C2-1031

1Z 1 -11 1 10 210
10 -2001

C3-1C2

Cli+2C2

R3+2R2

Ri;-1R2

11 121
-1u21

18-1;-11

111
1-12 Z -12-2 1 r 11 1121

17t111

18-U-11

R3+IRli111

1-12 z ,-12-2 1 Lm CU Rli 7C3 1R3 11 121
1111
1-- -U-2-1 11-6 1 -1 -1 2 91
-16 Z. 1 L_^* whichgives ,1 I* 11-6 P=" 21
111Q-
1 -1 -1 2 9 -16 -ij-2-11

Therefore

10 -1AQP 101-6
1-19

0-12-16

1 1 2 -ii 1 11 -2-1 2513
•31;-18 6233
-h-2 71
-10-1 182
-1 stillbecalculatedby -1A-QP.

However,sinceQisequaltoI,

AtheinverseofAisgivenby

-1 Q. itisnotnecessarytoformtheproductQP, n

METHODSINVOLVINGTHEDETERMINANTOFAMATRIX

A•adjA-adjA•A-|A|I.

If|A|?^0,thenthiseq^uationcanbewrittenas

adl_AadiA ^|A||A|^^'

Therefore,bydefinitionofthematrixinverse

_1adjA^|A| 12 then

Therefore

A ^1^22 ^2"^21 -^^11 adjA=C^2"^ "^21^1 23

25-3U62

13-1833

7-10181-12

-U -2 -1 idjA

251371

•3U-18-10-1

6233182

-h-2-1 withA,Forthisexamplethisgives

A'adJA

1-1 -21; -h 21
-1-1 3 -23 I

251371

-31-18-10'-1

6233182

-li-2'-1 10 10 10 1

J•

ofA•adjA.Inthisexanple, lAl=1. Hence

A•»adjA.

Hi |rl-A |rl-A| r-11+1-1

2r-U-3

1;r+2-3

-2-1T'h 0.

Evaluationofthisdeterminantgives

|rl-At"r^-7r^+l8r^-2$r+1=

A^-7A^+18A^-25a+I=Z

A^-7A^+18A^-25a»-I.

.(A^-7A^+18a-25l)A--I 'y-'i-f! 25
then (-A-^+7A^-18a+251)A=I. ofAisbydefixu-tion: a"-'-=(-A^+71?-18a+251) add. A-1 11 -36 -21; 93
-13-12 -k13 -26-8- 39
78
111
•51 7-Hi -70li2 98-35
56U2
-m-56* 5677
-56Ii2 798
-181818

36-72-5U

7236
-36-18 18 -72 25
25
25
25
,-1

251371

•3U-18-10-1

6233182

-1;-2-1 16 completed. characteristicequation.

MatrixInversionUsingtheTraceofaMatrix

thismethodcouldbeusedtoadvantage. ingseriesbeequaltoSj^: ^k-^l*^2*---^^n. 17

Sk=T(A^),

Pi'^1

2P2-S2-P^s^

^k"^kPl\-1-•••-Pk-1^1 ^Pn=^n-Pl^n-1Pn-1^1 computethepo>;ersofA. followingformulae:

C]_-T(AAq)A^-AAq-C3_I

Cg-1/2T(AA^)Ag=AA^-Cgl

18 <^=1/kT(AAj^_i)Aj,=AAi^_-L-Cj^I c^=1/nT(AAn_i)An»AA^.i-c^I whereA-isdefinedtobeI.Itwillbeshownthat ^k°Pkk»1,2,...,n andthat An=Z.

IfAisnonsingularandCj^j^0,then

^-1=^n^

Therefore

A"=An_i/cn.

Firstofallbychoice:

ci-T(AAq)=T(AI)=T(A)-SI-pi 19 also 22

AA-j_"A(AAq-c^I)=A-c-j^A=A-p-^A

2c2=T(aA3_)-T(a)-PiT(A)-Sg-P^s^-2p2.

Therefore

Cg"P2'

Therefore

3"P3 "^k+l-Pl^k-...-Pk^i-(k+l)Pk+i.

Therefore

°k+l"Pk+1*

Alsobytheinduction

A^=A-P3_I

2AAi-P2I=A-p^A-P2I

^n"^n-1-Pn^"^"^"^1^'"'^-...-P^I.

BytheHamilton-Cayleytheoremthen,

^l 20 puting ci=1/1T(AAq)=7 hAA^-71 -6-1-1-1 -2-33 -k-93 210-3

Cg=1/2T(AA^)=-18

12$S-1

A2-AA^+181-h

11123
-13 2U 11 -15quotesdbs_dbs20.pdfusesText_26

[PDF] [PDF] Numerical methods of matrix inversion - CORE

NUMERICALMETHODSOFMATRIXINVERSION

KEITHDONALDRUSH

B.S.,KansasStateUniversity,1962

AMASTER'SREPORT

MASTEROFSCIENCE

DepartmentofMathematics

KANSASSTATEUNIVERSITY

Manhattan,Kansas

Approvo^dby:

Majoi:'Professor

INVERSIONOFAMATRIXBYPARTITIONING..

TheInversionofaMatrixbyBordering28

ITERATIVEMETHODSOFMATRKINVERSION36

CONCLUSIONU6

ACKNOVTLEDGEIffiNTIi7

REFERENCES.ii8

INTRODUCTION

A-(a^g).

A+B-(a^s+bj.s).

A.B-AB-(21arkbks)k=l

AA-1»A"^A=I.

Thedeterminantofa2x2matrixisdefinedtobe

IX-A'-'-G

X-A'-'-G.

PAQ=I,then

QPAQ=Q.

QPA=I.

Thereforebydefinitionofmatrixinverse

Firstfom

1-1-1-1I1

21UJOOOI

Nowusingelementaryoperations

C2+ICl

C3+ICl1111

CU+ICl

R2+2R11

Rli-2R11

121011-2

I1-2-23

C2-1031

C3-1C2

Cli+2C2

R3+2R2

Ri;-1R2

18-1;-11

17t111

18-U-11

R3+IRli111

Therefore

0-12-16

However,sinceQisequaltoI,

AtheinverseofAisgivenby

METHODSINVOLVINGTHEDETERMINANTOFAMATRIX

A•adjA-adjA•A-|A|I.

If|A|?^0,thenthiseq^uationcanbewrittenas

Therefore,bydefinitionofthematrixinverse

Therefore

25-3U62

13-1833

7-10181-12

251371

6233182

A'adJA

251371

6233182

J•

A•»adjA.

2r-U-3

1;r+2-3

Evaluationofthisdeterminantgives

A^-7A^+18A^-25a+I=Z

A^-7A^+18A^-25a»-I.

36-72-5U

251371

6233182

MatrixInversionUsingtheTraceofaMatrix

Sk=T(A^),

2P2-S2-P^s^

C]_-T(AAq)A^-AAq-C3_I