Linear Algebra Done Right · Problem 7F.15

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · polar-decomposition

Problem

Define

T \in \mathcal{L}(\mathbf{F}^3)

T(z_1, z_2, z_3) = (z_3, 2z_1, 3z_2).

Find (explicitly) a unitary operator

S \in \mathcal{L}(\mathbf{F}^3)

such that

T = S\sqrt{T^*T}