Linear Algebra Done Right · Problem 7B.17

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · spectral-theorem

Problem

Suppose

\mathbf{F} = \mathbf{R}

and

\mathcal{E} \subseteq \mathcal{L}(V)

. Prove that there is an orthonormal basis of

V

with respect to which every element of

\mathcal{E}

has a diagonal matrix if and only if

S

and

T

are commuting self-adjoint operators for all

S, T \in \mathcal{E}

. This exercise extends the real spectral theorem to the context of a collection of commuting self-adjoint operators.