21.1. REVIEWOFVECTORSANDMATRICES 311
Therefore
λ∗nu(n)·u(m)=λmu(n)·u(m) (21.42)
Forn=m,thisimplies
λn isrealforalln (21.43)
whichprovesthefirstpartoftheorem.Forn+=m,andλn+=λm,italsoimplies
u(n)·u(m)= 0 λn+=λm (21.44)whichprovesthesecondpartoftheorem.
- Example LetsfindtheeigenvaluesandeigenvectorsoftheHermitianmatrix
H=
[
0 i
−i 0]
(21.45)First,tofindtheeigenvalues
det[
−λ i
−i −λ]
=λ^2 − 1 = 0 (21.46)sotheeigenvaluesare
λ 1 =+1 λ 2 =− 1 (21.47)Notethatthesearebothreal,inagreementwiththeoremabove. Thensolveforthe
eigenvectoru 1
[
0 i
−i 0][
u^11
u^12]
=[
u^11
u^12][
iu^12
−iu^11]
=[
u^11
u^12]
(21.48)whichhasthesolution
u^12 =−iu^11 or u^1 =[
u^11
−iu^11]
(21.49)Finally,wedetermineu^11 bynormalization
1 = u^1 ·u^1 =|u^11 |^2 +|−iu^11 |^2 = 2 |u^11 |^2=⇒ u 1 =