Lyapunov equation

Summary

The Lyapunov equation is the linear matrix equation $AX + XA^* + Q = 0$, or its discrete form $AXA^* - X + Q = 0$, for an unknown matrix $X$. Its solutions give quadratic Lyapunov functions for the linear system $x' = A^* x$ or $x_{k+1} = A^* x_k$. Every eigenvalue of $A$ has negative real part exactly when the continuous equation with a positive definite $Q$ has a positive definite solution. So the equation turns Lyapunov stability into linear algebra. The continuous form is a special case of the Sylvester equation.