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.