Linear Algebra Done Right · Problem 5E.9

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · commuting-operators

Problem

Suppose

V

is a finite-dimensional nonzero complex vector space. Suppose that

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

is such that

S

and

T

commute for all

S, T \in \mathcal{E}

. ((a)) Prove that there is a vector in

V

that is an eigenvector for every element of

\mathcal{E}

. ((b)) Prove that there is a basis of

V

with respect to which every element of

\mathcal{E}

has an upper-triangular matrix. This exercise extends 5.78 and 5.80, which consider 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.