Symmetric part of a matrix ($\mathrm{sym}A=(A+A^T)/2$)

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.