Linear Algebra Done Right · Problem 8A.12

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · generalized-eigenvectors-nilpotent

Problem

Suppose

T \in \mathcal{L}(V)

is such that every nonzero vector in

V

is a generalized eigenvector of

T

. Prove that there exists

\lambda \in \mathbf{F}

such that

T - \lambda I

is nilpotent.