Honors A: Fall 2013

Proof. We prove the statement by induction on n. n = 1 case can be easily verified. Let’s assume the theorem is true for n − 1. We take an orthonormal basis [e1, · · · , en] for W and write Lei = ∑n j=1 ajiej. Since L is symmetric, we have aji = aij or A = [aij] is a symmetric matrix. From the lemma, any root of the characteristic polynomial of A is real. Take λ1 to be a root, then A−λ1I is singular. Therefore, the linear transformation L − λ1I is singular, and there exists a u1 ∈ W such that Lu1 = λ1u1. Now we look at the orthogonal complement S of u1 in W with respect to 〈〈, 〉〉. It can be shown that, as in the proof of the spectral theorem, L maps S into itself. We can then apply the induction hypothesis to S which is an n− 1 dimensional space.