Linear Algebra Done Right · Problem 9C.19

Operators on Real Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · determinants

Problem

Suppose

V

is an inner product space,

e_1, \ldots, e_n

is an orthonormal basis of

V

, and

T \in \mathcal{L}(V)

is a positive operator. (a) Prove that

\det T \leq \prod_{k=1}^n \langle Te_k, e_k \rangle

. (b) Prove that if

T

is invertible, then the inequality in (a) is an equality if and only if

e_k

is an eigenvector of

T

for each

k = 1, \ldots, n