# Symmetric matrix

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A square matrix in which any two elements symmetrically positioned with respect to the main diagonal are equal to each other, that is, a matrix that is equal to its transpose: A real symmetric matrix of order has exactly real eigenvalues (counted with multiplicity). If is a symmetric matrix, then so are and , and if and are symmetric matrices of the same order, then is a symmetric matrix, while is symmetric if and only if .

Every square complex matrix is similar to a symmetric matrix. A real -matrix is symmetric if and only if the associated operator (with respect to the standard basis) is self-adjoint (with respect to the standard inner product). A polar decomposition factors a matrix into a product of a symmetric and an orthogonal matrix.
Let be a bilinear form on a vector space (cf. Bilinear mapping). Then the matrix of (with respect to the same basis in the two factors ) is symmetric if and only if is a symmetric bilinear form, i.e. .