Linear Algebra Done Right · Problem 7B.14

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · spectral-theorem

Problem

Suppose

\mathbf{F} = \mathbf{R}

and

T \in \mathcal{L}(V)

. Prove that

T

is self-adjoint if and only if all pairs of eigenvectors corresponding to distinct eigenvalues of

T

are orthogonal and

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

, where

\lambda_1, \ldots, \lambda_m

denote the distinct eigenvalues of

T