AdvancedSignalandImageProcessingProfessor:A.Mohammad–DjafariExercisenumber4:LinearAlgbera,Generalizedinversion,FacorAnalysisandSourcesseparationPartaA:LinearAlgebra,Generalizedinversion,MinimumNormeLeastSquaresinversion1.Weknowthesumg1andthedifferenceg2oftwonumbersf1andf2.Findthosenumbers.′′Help:Writetheproblemintheformofg=Afwithg=[g1,g2];f=[f1,f2]and11A=.Checkthenifthismatrixisinvertibleandinverseitandfindthe1−1−1solutionf=Ag.Fornumericalapplication,g1=10,g2=6.ba11a122.Weknowthatgisalinearfunctionoff,i.e.g=AfwithA=.a21a22•Showthatthisequationcanalsobewrittenintheformg=Fawitha=′[a11,a21,a12,a22].GivetheexpressionofF.•Iffandgaregiven,canwedetermineaorequivalentlyA?Why?•Ifwefixethevaluesofa11=1anda22=1,canwedeterminea12anda21?Givetheirexpressionsasafunctionofgandf.•Ifbetweenallthepossiblesolutions,wedecidetochoosetheonewithminimumnormekAk2=kak2,canyougivetheexpressionofthissolution?cos(θ)sin(θ)•IfA=isarotationmatrix.Canwedetermineθ?−sin(θ)sin(θ)a110cos(θ)sin(θ)•IfA=.Canwedetermineθ,a11anda22?If0a22−sin(θ)sin(θ)weimposea11=a22=a,canwedetermineaandθ?•Now,ifareonlygiveng,canwedetermineAandf?Isthesolutionunique?WhatifweimposekAk2tobeminimum,finditandthenuseittofindf?•Now,considerg(t)=Af(t),t=1,···,T.DefinethematricesG=[g(1);···;g(T)],F=[f(1);···;f(T)]andshowthatwecanwriteG=AF.IfTisgreatenough,canwedetermineAfromFandG?3.Considernowthegeneralcaseofg=AfwherethematrixAhasdimensions(M×N).•GivenAandgproposeasolution(exactorapproximate)forfforthefollowingcasesM=N,M<NandM>N.•Givenfandgproposeasolution(exactorapproximate)forAforthefollowingcasesM=N,M<NandM>N.Hasthisproblemanuniquesolution?CanweimposesomeconstraintsonelementsofAtobeabletofindauniquesolution?Whichone?1•IfweimposeminimumnormekAk2=kak2,canyougivetheexpressionofthissolution?4.Considernowthefollowingrelationsg=Afandf=BgwhereAandBaretwoinvertiblematricesofdimensions(N×N).Interpretethefollowingsituationsb•A=B=I.•ThejthcolumnofAisallzero.•ThejthcolumnofBis