Linear Algebra Done Right · Problem 7E.13

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · singular-value-decomposition

Problem

Suppose

T_1, T_2 \in \mathcal{L}(V)

. Prove that

T_1

and

T_2

have the same singular values if and only if there exist unitary operators

S_1, S_2 \in \mathcal{L}(V)

such that

T_1 = S_1 T_2 S_2