Definition of normal matrix

Definition. A complex square matrix $A$ is normal if it commutes with its conjugate transpose:
$$A A^* = A^* A$$

Examples of normal matrices:

• Hermitian matrices: $A = A^*$, so $AA^* = A^2 = A^*A$
• Skew-Hermitian matrices: $A = -A^*$
• Unitary matrices: $A^* = A^{-1}$, so $AA^* = I = A^*A$
• Diagonal matrices: they commute with everything

Example. $A = \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}$ is normal since $AA^T = A^T A = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$.

Normal matrices form the broadest class of matrices that can be unitarily diagonalized.