A Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose.
In linear algebra, a hermitian matrix is a square matrix that is equal to its conjugate transpose; this is occasionally known as the adjoint matrix as well. Formally, matrix $A$ is symmetric if $A^H=A$. Real symmetric matrices are de facto hermitian.
The sum and difference of two hermitian matrices is again hermitian, but this is not always true for the product: given hermitian matrices $A$ and $B$, then $AB$ is symmetric if and only if $A$ and $B$ commute, i.e., if $AB = BA$. So for integer $n$, $A^n$ is hermitian if $A$ is hermitian. If $A^{−1}$ exists, it is hermitian if and only if $A$ is hermitian.
An important fact about hermitian (symmetric) matrices is that they possess real eigenvalues. This can be seen by considering a finite-dimensional complex vector space equipped with the usual inner product.
