# Symmetric matrix

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

[itex]A^T = A[itex]

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:

[itex]\begin{bmatrix}

1 & 2 & 3\\ 2 & -4 & 5\\ 3 & 5 & 6\end{bmatrix}[itex]

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.

Other types of symmetry or pattern in square matrices have special names: see for example:

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