Trapezoidal contractivity for a semidiscrete heat equation ($\Vert U^{k+1}{\Vert }_M\le \Vert U^k{\Vert }_M$)

If $M$ is symmetric positive definite and $K$ symmetric negative semidefinite, the update $(M-\Delta tK/2)U^{k+1}=(M+\Delta tK/2)U^k$ is uniquely solvable for every positive step. Multiply its difference form by $(U^{k+1}+U^k{)}^T$ to obtainFor finite element heat flow, the mass norm is precisely the spatial $L^2$ norm, giving unconditional stability of the Crank-Nicolson method.