Definition of Hermitian/self-adjoint matrix

Definition. A complex square matrix $A$ is Hermitian (or self-adjoint) if it equals its conjugate transpose:
$$A = A^* \quad \text{where} \quad A^* = \overline{A}^T$$

Equivalently, $a_{ij} = \overline{a_{ji}}$ for all $i, j$. This means:

• Diagonal entries are real: $a_{ii} \in \mathbb{R}$
• Off-diagonal entries are complex conjugates of each other

Example. $A = \begin{pmatrix} 2 & 1+i \\ 1-i & 3 \end{pmatrix}$ is Hermitian.

A real Hermitian matrix is simply a real symmetric matrix. Hermitian matrices are the complex analogue of symmetric matrices.