In linear algebra, a symmetric matrix is a matrix that is its own transpose. Thus A is symmetric if:
which implies that A is a square matrix.
Intuitively, the entries of a symmetric matrix are symmetric with respect to the main diagonal (top left to bottom right). Example:
Any diagonal matrix is symmetric, since all its off-diagonal entries are zero.
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.
See also skew-symmetric matrix.
Other types of symmetry or pattern in square matrices have special names: see for example:
