Symmetric and antisymmetric parts of a matrix

Every square matrix decomposes uniquely as $A=(A+A^T)/2+(A-A^T)/2$, a symmetric part plus an antisymmetric part.

The symmetric part of a real square matrix is $H=(A+A^T)/2$. It controls the quadratic form because $x^{T}Ax=x^{T}Hx$; the antisymmetric part contributes zero. In a linear differential equation $x^{\prime }=Ax$, it determines the instantaneous derivative of the squared Euclidean norm.