Hermitian matrices

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.