Self-adjoint linear operator - Imported Knowlpedia Content

Let $H$ be an inner product space (real or complex), and let $A:H\to H$ be a linear operator.

The operator $A$ is self-adjoint if

\langle Ax,y\rangle=\langle x,Ay\rangle \quad \text{for all }x,y\in H.

Examples

In $\mathbb{R}^n$ with the standard inner product, $A(x)=Mx$ is self-adjoint iff $M$ is symmetric.
In $\mathbb{C}^n$ , $A(x)=Mx$ is self-adjoint iff $M$ is Hermitian.

Remarks

Context. Self-adjoint operators generalize symmetric (real) and Hermitian (complex) matrices and play a central role in convexity of quadratic forms and spectral theory.