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  "body": "\n## Summary\n\nThe Stein equation is the linear matrix equation $X - AXA^* = Q$ for an unknown square matrix $X$.\nIt is the discrete-time form of the [[Lyapunov equation|92GVPZH9FT]].\nWhen every eigenvalue of $A$ has absolute value less than $1$, it has the unique solution $X = \\sum_{k \\ge 0} A^k Q (A^*)^k$.\nThis links the stability of the recursion $x_{k+1} = A^* x_k$ with positive definite solutions of a linear equation.\n\n## Explanations\n\n### L1 human\n\nConsider a system that updates its state by a matrix at each time step.\nThe Stein equation asks for a quadratic energy that drops by a prescribed amount $Q$ at each step.\nFor one variable with $A = 1/2$ and $Q = 3$, the equation $X - X/4 = 3$ gives $X = 4$.\nThe solution adds up the contributions of $Q$ over all future steps, which is finite exactly when the system decays.\n",
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