Definition: A complex square matrix $A$ is normal if it commutes with its conjugate transpose:
$$AA^* = A^*A$$
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 ($A^T = A^{-1}$)
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.
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 $x$
Normal matrices generalize Hermitian and unitary matrices, capturing exactly those matrices that can be diagonalized by a unitary transformation.
**Definition:** A complex square matrix $A$ is **normal** if it commutes with its conjugate transpose:
$$AA^* = A^*A$$
**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 ($A^T = A^{-1}$)
**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.
**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 $x$
Normal matrices generalize Hermitian and unitary matrices, capturing exactly those matrices that can be diagonalized by a unitary transformation.