Stein equation

Remarks

It is the discrete Lyapunov equation $AXA^* - X + Q = 0$. More generally, the name is used for $X - AXB = C$ with given $A$, $B$, $C$. This equation has a unique solution exactly when $\lambda \mu \ne 1$ for every eigenvalue $\lambda$ of $A$ and every eigenvalue $\mu$ of $B$. If the spectral radius of $A$ is less than $1$, the unique solution is $X = \sum_{k=0}^{\infty} A^k Q (A^*)^k$, and $X$ is positive definite when $Q$ is. Example: for $n = 1$, $A = 1/2$ and $Q = 3$, the equation reads $X - X/4 = 3$, so $X = 4$; the series gives $3 \sum_{k \ge 0} 4^{-k} = 4$ as well. Non-example of solvability: for $n = 1$, $A = 1$ and $Q = 1$, the equation reads $0 = 1$ and has no solution, since $\lambda \bar{\lambda} = 1$ for the eigenvalue $\lambda = 1$.