Linear Algebra Done Right · Problem 7E.12

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · singular-value-decomposition

Problem

(a) Give an example of a finite-dimensional vector space and an operator

T

on it such that the singular values of

T^2

do not equal the squares of the singular values of

T

. (b) Suppose

T \in \mathcal{L}(V)

is normal. Prove that the singular values of

T^2

equal the squares of the singular values of

T