Linear Algebra Done Right · Problem 7F.2

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · operator-norms

Problem

Suppose that

T \in \mathcal{L}(V)

is self-adjoint or that

\mathbf{F} = \mathbf{C}

and

T \in \mathcal{L}(V)

is normal. Prove that

\|T\| = \max\{|\lambda| : \lambda \text{ is an eigenvalue of } T\}.