Linear Algebra Done Right · Problem 7C.5

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · positive-operators

Problem

Suppose

T \in \mathcal{L}(V)

is self-adjoint. Prove that

T

is a positive operator if and only if for every orthonormal basis

e_1, \ldots, e_n

V

, all entries on the diagonal of

\mathcal{M}(T, (e_1, \ldots, e_n))

are nonnegative numbers.