Linear Algebra Done Right · Problem 8B.18

Operators on Complex Vector Spaces — Sheldon Axler

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

Problem

Suppose

T \in \mathcal{L}(V)

and

\lambda

is an eigenvalue of

T

. Explain why the following four numbers equal each other. (a) The exponent of

z - \lambda

in the factorization of the minimal polynomial of

T

. (b) The smallest positive integer

m

such that

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

. (c) The smallest positive integer

m

such that

\operatorname{null}(T - \lambda I)^m = \operatorname{null}(T - \lambda I)^{m+1}

. (d) The smallest positive integer

m

such that

\operatorname{range}(T - \lambda I)^m = \operatorname{range}(T - \lambda I)^{m+1}