Linear Algebra Done Right · Problem 6A.19

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · inner-products-and-norms

Problem

Suppose

v_1, \ldots, v_n

is a basis of

V

and

T \in \mathcal{L}(V)

. Prove that if

\lambda

is an eigenvalue of

T

, then

|\lambda|^2 \leq \sum_{j=1}^{n} \sum_{k=1}^{n} |\mathcal{M}(T)_{j,k}|^2,

where

\mathcal{M}(T)_{j,k}

denotes the entry in row

j

, column

k

of the matrix of

T

with respect to the basis

v_1, \ldots, v_n