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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $A$ be an $n \\times n$ complex [[matrix|0JXZKX526R]], and let $Q$ be an $n \\times n$ [[Hermitian matrix|0P17NGQQMQ]].\nWrite $A^*$ for the [[conjugate transpose|0DXKE10PHF]] of $A$.\nThe *Stein equation* for $A$ and $Q$ is\n$$X - A X A^* = Q$$\nfor an unknown $n \\times n$ complex matrix $X$.\n\n## Remarks\n\nIt is the discrete [[Lyapunov equation|92GVPZH9FT]] $AXA^* - X + Q = 0$.\nMore generally, the name is used for $X - AXB = C$ with given $A$, $B$, $C$.\nThis equation has a unique solution exactly when $\\lambda \\mu \\ne 1$ for every eigenvalue $\\lambda$ of $A$ and every eigenvalue $\\mu$ of $B$.\nIf 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.\nExample: 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.\nNon-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 $\\lambda = 1$.\n",
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