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 $A=\|a_{ik}\|_1^n$ that is equal to its transpose:

$$a_{ik}=a_{ki},\quad i,k=1,\dots,n.$$

A real symmetric matrix of order $n$ has exactly $n$ real eigenvalues (counted with multiplicity). If $A$ is a symmetric matrix, then so are $A^{-1}$ and $A^p$, and if $A$ and $B$ are symmetric matrices of the same order, then $A+B$ is a symmetric matrix, while $AB$ is symmetric if and only if $AB=BA$.

Comments

Every square complex matrix is similar to a symmetric matrix. A real $(n\times n)$-matrix is symmetric if and only if the associated operator $\mathbf R^n\to\mathbf R^n$ (with respect to the standard basis) is self-adjoint (with respect to the standard inner product). A polar decomposition factors a matrix $A$ into a product $SQ$ of a symmetric and an orthogonal matrix.

Let $B\colon V\times V\to k$ be a bilinear form on a vector space $V$ (cf. Bilinear mapping). Then the matrix of $B$ (with respect to the same basis in the two factors $V$) is symmetric if and only if $B$ is a symmetric bilinear form, i.e. $B(u,v)=B(v,u)$.

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