Symmetric matrix

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 .

Comments

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. .

References

[a1]	F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1959–1960) pp. Vol. 1, Chapt. IX; Vol. 2, Chapt. XI (Translated from Russian)
[a2]	W. Noll, "Finite dimensional spaces" , M. Nijhoff (1987) pp. Sect. 2.7