Linear Algebra Done Right · Problem 5D.16

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · diagonalizable-operators

Problem

Suppose that

T \in \mathcal{L}(V)

is diagonalizable. Let

\lambda_1, \ldots, \lambda_m

denote the distinct eigenvalues of

T

. Prove that a subspace

U

V

is invariant under

T

if and only if there exist subspaces

U_1, \ldots, U_m

V

such that

U_k \subseteq E(\lambda_k, T)

for each

k

and

U = U_1 \oplus \cdots \oplus U_m