Linear Algebra Done Right · Problem 5E.2

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · commuting-operators

Problem

Suppose

\mathcal{E}

is a subset of

\mathcal{L}(V)

and every element of

\mathcal{E}

is diagonalizable. Prove that there exists a basis of

V

with respect to which every element of

\mathcal{E}

has a diagonal matrix if and only if every pair of elements of

\mathcal{E}

commutes. This exercise extends 5.76, which considers the case in which

\mathcal{E}

contains only two elements. For this exercise,

\mathcal{E}

may contain any number of elements, and

\mathcal{E}

may even be an infinite set.