Linear Algebra Done Right · Problem 7F.29

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · polar-decomposition

Problem

Suppose

T \in \mathcal{L}(V)

. (a) Use the polar decomposition to show that there exists a unitary operator

S \in \mathcal{L}(V)

such that

TT^* = ST^*TS^*

. (b) Show how (a) implies that

T

and

T^*

have the same singular values.