Linear Algebra Done Right · Problem 7E.11

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · singular-value-decomposition

Problem

Suppose that

T \in \mathcal{L}(V, W)

and

v_1, \ldots, v_n

is an orthonormal basis of

V

. Let

s_1, \ldots, s_n

denote the singular values of

T

. (a) Prove that

\|Tv_1\|^2 + \cdots + \|Tv_n\|^2 = s_1^2 + \cdots + s_n^2

. (b) Prove that if

W = V

and

T

is a positive operator, then

\langle Tv_1, v_1 \rangle + \cdots + \langle Tv_n, v_n \rangle = s_1 + \cdots + s_n.

See the comment after Exercise 5 in Section 7A.