Linear Algebra Done Right · Problem 8B.6

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · generalized-eigenspaces-minimal-polynomial

Problem

Suppose

T \in \mathcal{L}(V)

and

\lambda

is an eigenvalue of

T

. Explain why the exponent of

z - \lambda

in the factorization of the minimal polynomial of

T

is the smallest positive integer

m

such that

(T - \lambda I)^m|_{G(\lambda, T)} = 0