Linear Algebra Done Right · Problem 7B.13

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · spectral-theorem

Problem

Without using the complex spectral theorem, use the version of Schur's theorem that applies to two commuting operators (take

\mathcal{E} = \{T, T^*\}

in Exercise 20 in Section 6B) to give a different proof that if

\mathbf{F} = \mathbf{C}

and

T \in \mathcal{L}(V)

is normal, then

T

has a diagonal matrix with respect to some orthonormal basis of

V