On the maximum possible negativity margin for the first derivative (first difference) of a quadratic Lyapunov function (Q1776234)

Consider the linear system \[ \dot{x} = Ax \] and the quadratic Lyapunov function \[ V(x) = x^TPx \] such that \(A\) is Hurwitz and \(P>0\). If the derivative of \(V\) along the system, i.e. \(W(x) = x^T(A^TP + PA)x\), is considered, it is stated that its maximal value on the level surface \(V(x) = V_0\) is not less than \(2(\max_i\{\Re\,\lambda_i\})V_0\) where \(\lambda_i\) are the eigenvalues of \(A\). A discrete-time analogue is also stated.