Linear Algebra Done Right · Problem 7F.25

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · polar-decomposition

Problem

Fix

u, x \in V

with

u \neq 0

. Define

T \in \mathcal{L}(V)

Tv = \langle v, u \rangle x

for every

v \in V

. Prove that

\sqrt{T^*T}\, v = \frac{\|x\|}{\|u\|} \langle v, u \rangle u

for every

v \in V