# Hermitian matrices

Put content here**Definition:** A complex square matrix $A$ is **Hermitian** if it equals its conjugate transpose:
⏎
$$A = A^* \quad \text{or equivalently} \quad a_{ij} = \overline{a_{ji}}$$
⏎
For real matrices, Hermitian reduces to symmetric.
⏎
**Example:**
$$A = \begin{pmatrix} 2 & 1+i \\ 1-i & 3 \end{pmatrix}$$
⏎
**Properties:**
- All eigenvalues are real
- Eigenvectors corresponding to distinct eigenvalues are orthogonal
- $A$ is unitarily diagonalizable: $A = U\Lambda U^*$ where $U$ is unitary
- $x^*Ax$ is real for all vectors $x$
- The sum of Hermitian matrices is Hermitian
- The product $AB$ is Hermitian iff $A$ and $B$ commute
⏎
**Applications:**
- Quantum mechanics: observables are represented by Hermitian operators
- Signal processing: covariance matrices
- Optimization: Hessian of real-valued functions
⏎
Hermitian matrices are the complex analog of real symmetric matrices.

# Parents

* Particular types of matrices