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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.