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# Normal matrices

Put content here**Definition:** A complex square matrix $A$ is **normal** if it commutes with its conjugate transpose:
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$$AA^* = A^*A$$
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**Example:** All of the following are normal matrices:
- Hermitian matrices ($A = A^*$)
- Skew-Hermitian matrices ($A = -A^*$)
- Unitary matrices ($A^* = A^{-1}$)
- Symmetric matrices ($A = A^T$)
- Orthogonal matrices ($A^T = A^{-1}$)
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**Spectral Theorem for Normal Matrices:** $A$ is normal if and only if $A$ is unitarily diagonalizable: $A = UDU^*$ where $D$ is diagonal and $U$ is unitary.
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**Properties:**
- Normal matrices have an orthonormal basis of eigenvectors
- The sum and product of commuting normal matrices is normal
- If $A$ is normal, then $\|Ax\| = \|A^*x\|$ for all $x$
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Normal matrices generalize Hermitian and unitary matrices, capturing exactly those matrices that can be diagonalized by a unitary transformation.

# Parents

* Particular types of matrices