Linear Algebra Done Right · Problem 8B.10

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · generalized-eigenspaces-inner-product

Problem

Suppose

V

is a complex inner product space,

e_1, \ldots, e_n

is an orthonormal basis of

V

, and

T \in \mathcal{L}(V)

. Let

\lambda_1, \ldots, \lambda_n

be the eigenvalues of

T

, each included as many times as its multiplicity. Prove that

|\lambda_1|^2 + \cdots + |\lambda_n|^2 \leq \|Te_1\|^2 + \cdots + \|Te_n\|^2.

See the comment after Exercise 5 in Section 7A.