Linear Algebra Done Right · Problem 7E.10

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · singular-value-decomposition

Problem

Suppose

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

has singular values

s_1, \ldots, s_n

. Prove that if

T

is an invertible linear map, then

T^{-1}

has singular values

\frac{1}{s_n}, \ldots, \frac{1}{s_1}.