Symmetric matrix - Knowlpedia

A square matrix $A$ is symmetric when $A=A^{\mathsf T}$ , equivalently $A_{ij}=A_{ji}$ for all indices. Here $A^{\mathsf T}$ is the transpose.

Symmetric part

Over a field of characteristic different from two, every square matrix has the decomposition

A=\frac{A+A^{\mathsf T}}2+\frac{A-A^{\mathsf T}}2.

The first term is symmetric and the second is skew-symmetric. For real vectors, the skew-symmetric part contributes zero to $x^{\mathsf T}Ax$ .

Real and complex conventions

A real symmetric matrix is self-adjoint for the Euclidean inner product. For a complex matrix, symmetry $A=A^{\mathsf T}$ differs from the Hermitian condition $A=A^*$ .