Skip to main content

A symmetric matrix is a square matrix that is equal to its transpose.

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix $A$ is symmetric if $A^T=A$.

The sum and difference of two symmetric matrices is again symmetric, but this is not always true for the product: given symmetric 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 symmetric if $A$ is symmetric. If $A^{−1}$ exists, it is symmetric if and only if $A$ is symmetric.

The complex generalization is a hermitian matrix, a square matrix equal to its conjugate transpose. This is often denoted $A=A^{H}$ or $A=\overline{A^T}$; see for more information.