Linear Algebra Done Right · Problem 7A.11

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · normal-operators

Problem

Define an operator

S \colon \mathbf{F}^2 \to \mathbf{F}^2

S(w, z) = (-z, w)

. (a) Find a formula for

S^*

. (b) Show that

S

is normal but not self-adjoint. (c) Find all eigenvalues of

S

. If

\mathbf{F} = \mathbf{R}

, then

S

is the operator on

\mathbf{R}^2

of counterclockwise rotation by

90^\circ