302 CHAPTER 12. SELF ADJOINT OPERATORS

Thus Re (Av1 − λ1v1, w) = 0 for all w ∈ X. This implies Av1 = λ1v1. To see this, let w ∈ Xbe arbitrary and let θ be a complex number with |θ| = 1 and

|(Av1 − λ1v1, w)| = θ (Av1 − λ1v1, w) .

Then|(Av1 − λ1v1, w)| = Re

(Av1 − λ1v1, θw

)= 0.

Since this holds for all w, Av1 = λ1v1.Continuing with the proof of the theorem, let X2 ≡ {v1}⊥ . This is a closed subspace of

X and A : X2 → X2 because for x ∈ X2,

(Ax, v1) = (x,Av1) = λ1 (x, v1) = 0.

Letλ2 ≡ inf {(Ax, x) : |x| = 1, x ∈ X2}

As before, there exists v2 ∈ X2 such that Av2 = λ2v2, λ1 ≤ λ2. Now let X3 ≡ {v1, v2}⊥

and continue in this way. As long as k < n, it will be the case that {v1, · · · , vk}⊥ ̸= {0}.This is because for k < n these vectors cannot be a spanning set and so there exists somew /∈ span (v1, · · · , vk) . Then letting z be the closest point to w from span (v1, · · · , vk) , itfollows that w − z ∈ {v1, · · · , vk}⊥. Thus there is an decreasing sequence of eigenvalues{λk}nk=1 and a corresponding sequence of eigenvectors, {v1, · · · , vn} with this being anorthonormal set. ■

Contained in the proof of this theorem is the following important corollary.

Corollary 12.3.3 Let A ∈ L (X,X) be self adjoint where X is a finite dimensional innerproduct space. Then all the eigenvalues are real and for λ1 ≤ λ2 ≤ · · · ≤ λn the eigenvaluesof A, there exists an orthonormal set of vectors {u1, · · · , un} for which

Auk = λkuk.

Furthermore,λk ≡ inf {(Ax, x) : |x| = 1, x ∈ Xk}

whereXk ≡ {u1, · · · , uk−1}⊥ , X1 ≡ X.

Corollary 12.3.4 Let A ∈ L (X,X) be self adjoint (Hermitian) where X is a finite dimen-sional inner product space. Then the largest eigenvalue of A is given by

max {(Ax,x) : |x| = 1} (12.5)

and the minimum eigenvalue of A is given by

min {(Ax,x) : |x| = 1} . (12.6)

Proof: The proof of this is just like the proof of Theorem 12.3.2. Simply replace infwith sup and obtain a decreasing list of eigenvalues. This establishes 12.5. The claim 12.6follows from Theorem 12.3.2. ■

Another important observation is found in the following corollary.

Corollary 12.3.5 Let A ∈ L (X,X) where A is self adjoint. Then A =∑

i λivi⊗vi whereAvi = λivi and {vi}ni=1 is an orthonormal basis.