Linear Algebra, Theory And Applications, 2012a

312 SELF ADJOINT OPERATORS
Definition 13.2.3 Let L ∈L(H, H) where H is a finite dimensional inner product space.
Then L is Hermitian if L ∗ = L.
Theorem 13.2.4 Let L ∈L(H, H) where H is an n dimensional inner product space. If
L is Hermitian, then all of its eigenvalues λ k are real and there exists an orthonormal basis
of eigenvectors {w k } such that
L = ∑ λ k w k ⊗w k .
k
Proof: By Schur’s theorem, Theorem 13.2.2, there exist l ij ∈ F such that
Then by Lemma 12.4.2,
n∑
j=1 i=1
L =
n∑
j=1 i=1
j∑
l ij w i ⊗w j = L = L ∗ =
=
n∑
j=1 i=1
j∑
l ij w i ⊗w j
n∑
j=1 i=1
j∑
l ij w j ⊗w i =
j∑
(l ij w i ⊗w j ) ∗
n∑
i=1 j=1
i∑
l ji w i ⊗w j
By independence, if i = j,
l ii = l ii
and so these are all real. If i