Symmetric matrix

In linear algebra, a symmetric matrix is a matrix that is its own transpose. Thus A is symmetric if:

${\displaystyle A^{T}=A}$

which implies that A is a square matrix.

Examples

The entries of a symmetric matrix are symmetric with respect to the main diagonal (top left to bottom right). Example:

${\displaystyle {\begin{bmatrix}1&2&3\\2&0&5\\3&5&6\end{bmatrix}}}$ 

Any diagonal matrix is symmetric, since all its off-diagonal entries are zero.

Properties

One of the basic theorems concerning such matrices is the finite-dimensional spectral theorem, which says that any symmetric matrix whose entries are real can be diagonalized by an orthogonal matrix. This is a special case of a Hermitian matrix.

See also skew-symmetric (or antisymmetric) matrix.

Other types of symmetry or pattern in square matrices have special names: see for example:

• circulant matrix
• Hankel matrix
• Toeplitz matrix.