Hermitian Matrix

• categories: Linear algebra, Matrix

Definition
A complex square Matrix $A\in {\mathbb{C}}^{n\times n}$ is called Hermitian if it is equal to its conjugate transpose, i.e.,
$A=A^H,\text{where }{A}^H=\overline{A^{\top }}.$
This means that $a_{ij}=\overline{a_{ji}}$ for all $i,j$, where $\overline{z}$ denotes the complex conjugate of $z$.

Intuition
A Hermitian matrix generalizes the concept of a Symmetric Matrix to the complex domain. Its elements are mirrored across the diagonal, but with complex conjugation applied.

Key Properties

1. Real Diagonal Entries: The diagonal entries of a Hermitian matrix are always real, as $a_{ii}=\overline{a_{ii}}$.
2. Complex Conjugate Symmetry: Off-diagonal entries satisfy $a_{ij}=\overline{a_{ji}}$.
3. Real Eigenvalues: All eigenvalues of a Hermitian matrix are real.
4. Orthogonal Eigenvectors: The eigenvectors corresponding to distinct eigenvalues of a Hermitian matrix are orthogonal in the sense of the complex inner product.
5. Unitary Diagonalization: A Hermitian matrix is diagonalizable by a unitary matrix, i.e., $A=U\Lambda U^H$, where $U$ is unitary ($U^{H}U=I$) and $\Lambda$ is diagonal.
6. Positive Semi-Definite Matrix: A Hermitian matrix is positive semidefinite if all its eigenvalues are non-negative.

Applications

• Quantum Mechanics: Hermitian matrices represent observables (e.g., operators corresponding to measurable quantities like energy).
• Signal processing: Covariance matrices in signal processing are often Hermitian.