Linear Algebra Done Right · Problem 8B.7

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · generalized-eigenspaces

Problem

Suppose

T \in \mathcal{L}(V)

and

\lambda

is an eigenvalue of

T

with multiplicity

d

. Prove that

G(\lambda, T) = \operatorname{null}(T - \lambda I)^d

. If

d < \dim V

, then this exercise improves 8.20.