## Encyclopedia > Symmetric matrix

Article Content

# Symmetric matrix

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

$A^T = A$

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:

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

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

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.

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Father Damien ... After twelve years of ministering to the patients at the leper colony (see Kalawao County, Hawaii), he contracted the disease, from which he died at the colony. He is ...