Linear Algebra Done Right · Problem 6C.12

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · orthogonal-projections

Problem

Suppose

V

is finite-dimensional,

T \in \mathcal{L}(V)

, and

U

is a subspace of

V

. Prove that

U \text{ and } U^\perp \text{ are both invariant under } T \iff P_U T = T P_U.