Linear Algebra Done Right · Problem 7E.2

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · singular-value-decomposition

Problem

Suppose

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

and

s > 0

. Prove that

s

is a singular value of

T

if and only if there exist nonzero vectors

v \in V

and

w \in W

such that

Tv = sw \quad \text{and} \quad T^*w = sv.

The vectors

v, w

satisfying both equations above are called a Schmidt pair. Erhard Schmidt introduced the concept of singular values in 1907.