Linear Algebra Done Right · Problem 7F.32

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · singular-value-decomposition-ellipsoid

Problem

Suppose that

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

and

T \neq 0

. Let

s_1, \ldots, s_m

denote the positive singular values of

T

. Show that there exists an orthonormal basis

e_1, \ldots, e_m

(\operatorname{null} T)^\perp

such that

T\!\left(E\!\left(\frac{e_1}{s_1}, \ldots, \frac{e_m}{s_m}\right)\right)

equals the ball in

\operatorname{range} T

of radius 1 centered at 0.