Linear Algebra Done Right · Problem 7A.10

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · self-adjoint-operators

Problem

Suppose

\mathbf{F} = \mathbf{C}

and

T \in \mathcal{L}(V)

. Prove that

T

is self-adjoint if and only if

\langle Tv, v \rangle = \langle T^*v, v \rangle

for all

v \in V