---
tag: W90XPR88E8
kind: definition
name: Stein equation
msc: [15A24, 39B42, 93D05]
defines: FM9Y4Y6D8F
uses: [0JXZKX526R, 0P17NGQQMQ, 0DXKE10PHF, 92GVPZH9FT, 9V7YQA85PC, 0KX66CZMBB]
sources:
  - url: https://en.wikipedia.org/wiki/Lyapunov_equation
---

## Definition

Let $n \ge 1$, let $A$ be an $n \times n$ complex [[matrix|0JXZKX526R]], and let $Q$ be an $n \times n$ [[Hermitian matrix|0P17NGQQMQ]].
Write $A^*$ for the [[conjugate transpose|0DXKE10PHF]] of $A$.
The *Stein equation* for $A$ and $Q$ is
$$X - A X A^* = Q$$
for an unknown $n \times n$ complex matrix $X$.

## Remarks

It is the discrete [[Lyapunov equation|92GVPZH9FT]] $AXA^* - X + Q = 0$.
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|9V7YQA85PC]] 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|0KX66CZMBB]] 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 $\lambda = 1$.