Linear Algebra Done Right · Problem 5D.19

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · diagonalizable-operators

Problem

Prove or give a counterexample: If

T \in \mathcal{L}(V)

and there exists a subspace

U

V

that is invariant under

T

such that

T|_U

and

T/U

are both diagonalizable, then

T

is diagonalizable. See Exercise 13 in Section 5C for an analogous statement about upper-triangular matrices.