Solving the discrete Lyapunov equation using the solution of the corresponding continuous Lyapunov equation and vice versa

It is demonstrated that for arbitrary linear autonomous systems, a quadratic form yielding a Lyapunov function for the continuous-time system can be used to derive rather easily a Lyapunov function of the same type for the corresponding discrete-time system resulting from it by sampling with sampling period τ. Conversely, if for an arbitrary discrete system allowing an interpretation as sampled-data system, i.e. having a regular system matrix F, a quadratic form yielding a Lyapunov function is found, then a Lyapunov function for the corresponding continuous-time system can be computed rather easily from it