Linear Algebra Done Right · Problem 8B.20

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · characteristic-polynomial-invariant-subspaces

Problem

Suppose that

\mathbf{F} = \mathbf{C}

and

V_1, \ldots, V_m

are nonzero subspaces of

V

such that

V = V_1 \oplus \cdots \oplus V_m.

Suppose

T \in \mathcal{L}(V)

and each

V_k

is invariant under

T

. For each

k

, let

p_k

denote the characteristic polynomial of

T|_{V_k}

. Prove that the characteristic polynomial of

T

equals

p_1 \cdots p_m