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LEMMA

†
†
(AB ) = B A † (8.73)
PROOF From the definition of matrix multiplication and Hermitian adjoint,
we have:

†
[(AB ) ] = (A B )
ij ji
ki ∑
= ∑ AB = (A † ) (B † ) ik
kj
jk
k k
= ∑ (B † ) (A † ) = (B A † ) ij
†
ik
kj
k
DEFINITION A matrix is Hermitian if it is equal to its Hermitian adjoint;
that is

†
H = H (8.74)

THEOREM 1
The eigenvalues of a Hermitian matrix are real.
PROOF Let λ be an eigenvalue of H and let v m be the corresponding
m
eigenvector; then:

H v =λ v (8.75)
m m m

Taking the Hermitian adjoints of both sides, using the above lemma, and
remembering that H is Hermitian, we successively obtain:

†
†
(H v ) = v H = v H = v λ (8.76)
m m m m m
Now multiply (in an inner-product sense) Eq. (8.75) on the left with the bra
v and Eq. (8.76) on the right by the ket-vector v , we obtain:
m m

v H v = λ v v = λ v v ⇒ λ = λ (8.77)
m m m m m m m m m m

THEOREM 2
The eigenvectors of a Hermitian matrix corresponding to different eigenvalues are
orthogonal; that is, given that:

© 2001 by CRC Press LLC