310 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA
or,inthecaseofaD×Dmatrix,solvetheDsimultaneousequations
∑D
j=1
miju(jn)=λnu(in) (i= 1 , 2 , 3 ,...,D) (21.34)
Thegeneralsolutionofthisequation,intermsofdeterminants,isgiveninanylinear
algebrabook.Thesolutionisdenotedu(n),andshouldbenormalizedsothat
[u∗ 1 ,u∗ 2 ]
[
u 1
u 2
]
=
∑D
i=1
u∗iui= 1 (21.35)
Theorem
Theeigenvalues ofaHermitianmatrixareallreal. Anytwoeigenvectorsof a
Hermitianmatrix,whichcorrespondtodifferenteigenvalues,areorthogonal.
Proof: Let u(m) andu(n) betwoeigenvectorsof a Hermitianmatrix H, with
eigenvaluesλn, λmrespectively. Considerthequantity
Q=
∑
ij
(uni)∗Hiju(jm) (21.36)
Sinceumisaneigenvector,weknowthat
∑
j
Hiju(jm)=λmu(im) (21.37)
so
Q=λm
∑
i
(u(in))∗u(im)=λmu(n)·u(m) (21.38)
Ontheotherhand,wecanwriteQas
Q=
∑
ij
(
Hij∗u(in)
)∗
u(jm) (21.39)
BecauseHisanHermitianmatrix,
Hij∗=Hji (21.40)
so
Q =
∑
ij
(
Hjiu(in)
)∗
u(jm)
=
∑
j
(λnu(jn))∗u(jm)
= λ∗nu(n)·u(m) (21.41)