Linear Algebra Done Right · Problem 8B.8

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · generalized-eigenspace-decomposition

Problem

Suppose

T \in \mathcal{L}(V)

and

\lambda_1, \ldots, \lambda_m

are the distinct eigenvalues of

T

. Prove that

V = G(\lambda_1, T) \oplus \cdots \oplus G(\lambda_m, T)

if and only if the minimal polynomial of

T

equals

(z - \lambda_1)^{k_1} \cdots (z - \lambda_m)^{k_m}

for some positive integers

k_1, \ldots, k_m

. The case

\mathbf{F} = \mathbf{C}

follows immediately from 5.27(b) and the generalized eigenspace decomposition (8.22); thus this exercise is interesting only when

\mathbf{F} = \mathbf{R}