Linear Algebra Done Right · Problem 7F.31

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · polar-decomposition-normal

Problem

Suppose

\mathbf{F} = \mathbf{C}

and

T \in \mathcal{L}(V)

is normal. Prove that there exists a unitary operator

S \in \mathcal{L}(V)

such that

T = S\sqrt{T^*T}

and such that

S

and

\sqrt{T^*T}

both have diagonal matrices with respect to the same orthonormal basis of

V