Self-adjoint linear operator

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

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

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.