矩阵分析讲义 Matrix Factorization.pdf

Leture6:MatrixFactorization

MatrixAnalysis@HITSZ

Instructor:ZijunLuo

Fall2012

Readingassignment

•Section3.5.1~3.5.5(LU)

•Section2.6.1~2.6.3(QR)

•Section7.3:(7),(8)(GeneralInverse,

LeastSquare)

Final

Finalexam:Nov.12th,Tue.18:00-20:00

T

Det(A*)=Det(A)?No.ButDet(A)=Det(A).

AlgorithmtofindJordancanonicalformofamatrix

Wedefinethefollowingspaces,whereAisour

m×mmatrix:

Null(A-I)=Nullspace(A-I).Thedimension

ofthisisthenumberofeigenvectorsfor.

V=W=Null(A-I)

11

i

V=Null((A-I))

i

i

W={wNull(((A-I))suchthatwNull((A-

i

I)i-1)},for2im.

AlgorithmtofindJCF

ThisLecturehighlightsafewofthemany

factorizationtheoremsformatrices.Mainly

includesPLUdecompositionandQR

factorizationwhichresultsservetosimplify

thesolutiontolinearsystems.

Moreovergeneralizedinverseisalso

introducedtoinvertanon-invertiblematrix

foranon-consistentsystem.

Inthislecture,withoutexceptionthe

underlyinginnerproductisstandardinner

productandtheunderlyingvectornormis

Euclideannorm||·||.

2

7.1PLUDecompositionandLU/LR

factorization

ToachievethePLUandLU/LRdecomposition(or

factorization)ofaninvertiblematrixA,werequirea

modifiednotionoftherowreducedechelonformwhile

westudiedGaussianelimination.

Definition7.1.1.Themodifiedrowechelonformofa

matrixisthatformwhichsatisfiesalltheconditionsofthe

rowreducedechelonformexceptthatwedonotrequi