For comparison, consider the one-dimensional case, where this just says that the solution of is . Relationship Between Discrete and Continuous Lyapunov Equations We start with the continuous-time linear dynamics: . And then discretize it as follows: Where indicates a small forward displacement in time. Substituting the bottom equation into the top and shuffling terms around, we get a discrete-time equation for . Where we've defined . Now we can use the discrete time Lyapunov equation for : Plugging in our definition for , we get: Expanding this expression out yields: Recall that is a small displacement in time. Letting go to zero brings us closer and closer to having continuous dynamics—and in the limit we achieve them. It stands to reason that we should also recover the continuous-time Lyapunov equations in the limit as well. Dividing through by on both sides, and then letting we find that: which is the continuous-time Lyapunov equation, as desired. See also Sylvester equation Algebraic Riccati equation Kalman filter References Control theory