Hermitian matrix

Definition

Let $A = (a_{ij})$ be an $n \times n$ complex square matrix. The matrix $A$ is Hermitian when $A^* = A$, where $A^*$ is the conjugate transpose of $A$. Equivalently, $a_{ij} = \overline{a_{ji}}$ for all $i$ and $j$.

Remarks

The diagonal entries of a Hermitian matrix are real. A real matrix is Hermitian if and only if it is symmetric. The eigenvalues of a Hermitian matrix are real, and $\C^n$ has an orthonormal basis of eigenvectors, by the spectral theorem for Hermitian matrices. The matrix $\begin{pmatrix} 2 & 1 - i \\ 1 + i & 3 \end{pmatrix}$ is Hermitian.

Explanations

L1 human

A Hermitian matrix is the complex version of a symmetric matrix. The entry across the diagonal is the complex conjugate of the entry here. In quantum mechanics, observable quantities are Hermitian matrices or operators.