Stein equation

Summary

The Stein equation is the linear matrix equation $X - AXA^* = Q$ for an unknown square matrix $X$. It is the discrete-time form of the Lyapunov equation. When every eigenvalue of $A$ has absolute value less than $1$, it has the unique solution $X = \sum_{k \ge 0} A^k Q (A^*)^k$. This links the stability of the recursion $x_{k+1} = A^* x_k$ with positive definite solutions of a linear equation.