Symmetric matrix
|
|
In linear algebra, a symmetric matrix is a matrix that is its own transpose. Thus A is symmetric if:
- <math>A^T = A<math>
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:
- <math>\begin{bmatrix}
1 & 2 & 3\\ 2 & -4 & 5\\ 3 & 5 & 6\end{bmatrix}<math>
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:
es:Matriz simétrica fr:Matrice symétrique it:Matrice simmetrica nl:Symmetrische matrix ja:対称行列