A self-adjoint matrix (or Hermitian matrix) is one in which the matrix, $A$, is equal to its conjugate transpose, $A^*$.

If $A=A^*$, then the matrix is self-adjoint (or Hermitian)

Where:

$A^*=$ Conjugate transpose of $A$

Given a complex number $a + ib$, the conjugate is $a-ib$.

Thus, the following matrix is an example of a self-adjoint matrix:

$[X]=\begin{bmatrix} 1 & a-ib & c-id \\ a+ib & 1 & e-if \\ c+id & e+if & 1 \end{bmatrix}$

When a matrix consists of real parts only, a self-adjoint matrix is one where $A=A^T$.