Linear Algebra Done Right · Problem 8A.4

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · generalized-eigenvectors

Problem

Suppose

T \in \mathcal{L}(V)

\lambda \in \mathbf{F}

, and

m

is a positive integer such that the minimal polynomial of

T

is a polynomial multiple of

(z - \lambda)^m

. Prove that

\dim \operatorname{null}(T - \lambda I)^m \geq m.