Self-Adjoint Operators

Theorems 1–2 of the LA-I symmetric-matrices note proved real eigenvalues and orthogonal eigenvectors for real symmetric matrices, working directly with

A^\top=A

. This note restates that condition at the operator level —

T^*=T

, self-adjointness — using the adjoint built in the previous note, and re-derives the same two facts with proofs that work over

\mathbb{C}

as well as

\mathbb{R}

, setting up the full spectral theorem in the next topic.

Learning objectives. By the end of this note you should be able to:

state the definition of a self-adjoint operator and its matrix characterisation;
prove a self-adjoint operator's eigenvalues are real, using the adjoint-based argument;
prove eigenvectors for distinct eigenvalues of a self-adjoint operator are orthogonal;
distinguish "self-adjoint" from "symmetric" and know when the two coincide.

In this lesson

The full lesson, with worked examples and exercise solutions, is for members.