Linear Algebra Done Right · Problem 5E.7

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · commuting-operators

Problem

Suppose

V

is a complex vector space,

S \in \mathcal{L}(V)

is diagonalizable, and

T \in \mathcal{L}(V)

commutes with

S

. Prove that there is a basis of

V

such that

S

has a diagonal matrix with respect to this basis and

T

has an upper-triangular matrix with respect to this basis.