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  "body": "\n## Summary\n\nThe Lyapunov equation is the linear matrix equation $AX + XA^* + Q = 0$, or its discrete form $AXA^* - X + Q = 0$, for an unknown matrix $X$.\nIts solutions give quadratic [[Lyapunov functions|01REYCJ371]] for the linear system $x' = A^* x$ or $x_{k+1} = A^* x_k$.\nEvery eigenvalue of $A$ has negative real part exactly when the continuous equation with a positive definite $Q$ has a positive definite solution.\nSo the equation turns [[Lyapunov stability|0F2SYCKDT2]] into linear algebra.\nThe continuous form is a special case of the [[Sylvester equation|M9SW4NVP2H]].\n\n## Explanations\n\n### L1 human\n\nTo show that a system comes to rest, one looks for an energy that always decreases along its motion.\nFor a linear system the natural energies are quadratic, $x^* X x$ for a matrix $X$.\nThe Lyapunov equation is the condition on $X$ that makes this energy decrease at a prescribed rate.\nFor one variable $x' = ax$ with $a \u003c 0$, the energy $x^2$ decreases, and the equation $2aX + q = 0$ has the positive solution $X = -q/(2a)$ for $q \u003e 0$.\n",
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