Symmetric matrix
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.