e ultiplicit 2 will sho ho to calculate the eigen alues and the asso ciated eigen ectors in this situation 1 First need to solv the secular equation to get eigen alues 1 2 1 Therefore the eigen alues of are 2 Then can 64257nd the corresp onding eigen ID: 74940
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MatrixwithDegenerateEigenvaluesHereisamatrixwhichhasanondegenerateeigenvalue(1=2)andtwodegenerateeigenvalues= 1(i.e.multiplicityg=2).A=0@0111011101AWewillshowhowtocalculatetheeigenvaluesandtheassociatedeigenvectorsinthissituation:1.First,weneedtosolvethesecularequationjA 1jtogeteigenvalues:jA 1j= 111 111 (1)= 3+2+3= ( 2)(+1)2=0Therefore,theeigenvaluesofAare= 1; 1;2:2.Then,wecan ndthecorrespondingeigenvectors:(a)=20@ 2111 2111 21A0@x1x2x31A=0Wegetthreeequations: 2x1+x2+x3=0(2)x1 2x2+x3=0(3)x1+x2 2x3=0(4)Onlytwoofthreeequationsarelinearlyindependent.Thesolutionisx1= 0@1111A:Thegeometricalinterpretationisthatanyvectorlyinginthissubspace(aline)isaneigenvectorwitheigenvalue=2,thoughtheyarealllinearlydepedent.Sinceweareonlyinterestedinthelinearlydependentsolution,itisconvenienttochoosethenormalizedsolution:x1=1p30@1111A(5)1 (b)= 1isadegenerateeigenvaluewithmultiplicityg=2.Substituethisvaluebacktothesecularequation,Eq.(1),weget0@1111111111A0@x1x2x31A=0:Duetothedegeneracyg=2,weonlyhaveoneequationx1+x2+x3=0leftforthreeunknowns!Anyvectorinthissubspace(aplanepassingthroughtheorigin)isaneigenvectorwitheigenvalue= 1.Wecanarbitrarilychoosetwolinearlyindependentvectorsinthissubspaceasx2=0@11 21Ax3=0@ 2111A:Notealsothatthesetwoeigenvectorsarelinearlyindependent,butnotorthogonaltoeachother.WecanmakethemorthonormalbyGram-Schmidtprocedure:y2=x2jx2j=1p60@11 21Ay03=x3 x2xT2xT2x2x3=0@ 2111A 0@11 21A 36=0@ 323201AAfternormalization:y3=y03jy03j=1p20@1 101AThereforewegettwoorthonormaleigenvectors[y2y3]belongingtothedegenerateeigenvalue= 1.However,theyarenottheonlyeigenvectorsassociatedwiththiseigenvalue,anylinearcombinationy=c2y2+c3y3isalsoaneigenvector.Besidesy2andy3,therearein nitewaystochoosethebasistorepresenttheeigensubspaceassociatedwith= 1.Inpresentsituation,theeigensubspaceisaplanepassingthroughtheorigin,butorthogonaltothelinede nedbyvectorxT1=[111].Ifweallowedcomplexvalues,anotherpossiblechoiceofbasisfortheeigenvectorsinthedegeneratemanifoldisx02=1p30@11Ax03=1p30@11A2