12.3. SPECTRAL THEORY OF SELF ADJOINT OPERATORS 301

Theorem 12.3.2 Let A ∈ L (X,X) be self adjoint (Hermitian) where X is a finite dimen-sional inner product space of dimension n. Thus A = A∗. Then there exists an orthonormalbasis of eigenvectors, {vj}nj=1 .

Proof: Consider (Ax, x) . This quantity is always a real number because

(Ax, x) = (x,Ax) = (x,A∗x) = (Ax, x)

thanks to the assumption that A is self adjoint. Now define

λ1 ≡ inf {(Ax, x) : |x| = 1, x ∈ X1 ≡ X} .

Claim: λ1 is finite and there exists v1 ∈ X with |v1| = 1 such that (Av1, v1) = λ1.Proof of claim: Let {uj}nj=1 be an orthonormal basis for X and for x ∈ X, let (x1, · · · ,

xn) be defined as the components of the vector x. Thus,

x =

n∑j=1

xjuj .

Since this is an orthonormal basis, it follows from the axioms of the inner product that

|x|2 =

n∑j=1

|xj |2 .

Thus

(Ax, x) =

 n∑k=1

xkAuk,∑j=1

xjuj

 =∑k,j

xkxj (Auk, uj) ,

a real valued continuous function of (x1, · · · , xn) which is defined on the compact set

K ≡ {(x1, · · · , xn) ∈ Fn :

n∑j=1

|xj |2 = 1}.

Therefore, it achieves its minimum from the extreme value theorem. Then define

v1 ≡n∑

j=1

xjuj

where (x1, · · · , xn) is the point of K at which the above function achieves its minimum.This proves the claim.

I claim that λ1 is an eigenvalue and v1 is an eigenvector. Letting w ∈ X1 ≡ X, thefunction of the real variable, t, given by

f (t) ≡ (A (v1 + tw) , v1 + tw)

|v1 + tw|2=

(Av1, v1) + 2tRe (Av1, w) + t2 (Aw,w)

|v1|2 + 2tRe (v1, w) + t2 |w|2

achieves its minimum when t = 0. Therefore, the derivative of this function evaluated att = 0 must equal zero. Using the quotient rule, this implies, since |v1| = 1 that

2Re (Av1, w) |v1|2 − 2Re (v1, w) (Av1, v1) = 2 (Re (Av1, w)− Re (v1, w)λ1) = 0.