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# Definition of normal matrix

Put content here**Definition.** A complex square matrix $A$ is *normal* if it commutes with its conjugate transpose:
$$A A^* = A^* A$$
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**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
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**Example.** $A = \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}$ is normal since $AA^T = A^T A = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$.
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Normal matrices form the broadest class of matrices that can be unitarily diagonalized.

# Parents

* Normal matrices