Linear Algebra Done Right · Problem 7E.4

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · singular-value-decomposition

Problem

Suppose that

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

s_1

is the largest singular value of

T

, and

s_n

is the smallest singular value of

T

. Prove that

\{\|Tv\| : v \in V \text{ and } \|v\| = 1\} = [s_n, s_1].