Linear Algebra Done Right · Problem 7C.20

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · positive-operators

Problem

Suppose

T \in \mathcal{L}(V)

and

e_1, \ldots, e_n

is an orthonormal basis of

V

. Prove that

T

is a positive operator if and only if there exist

v_1, \ldots, v_n \in V

such that

\langle Te_k, e_j \rangle = \langle v_k, v_j \rangle

for all

j, k = 1, \ldots, n

. The numbers

\{\langle Te_k, e_j \rangle\}_{j,k=1,\ldots,n}

are the entries in the matrix of

T

with respect to the orthonormal basis

e_1, \ldots, e_n