Linear Algebra Done Right · Problem 20

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · upper-triangular-matrices

Problem

Suppose

\mathbf{F} = \mathbf{C}

V

is finite-dimensional, and

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

is such that

ST = TS

for all

S, T \in \mathcal{E}

. Prove that there is an orthonormal basis of

V

with respect to which every element of

\mathcal{E}

has an upper-triangular matrix. This exercise strengthens Exercise 9(b) in Section 5E (in the context of inner product spaces) by asserting that the basis in that exercise can be chosen to be orthonormal.

Browse the problem library