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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 $A$.\n\n1. The *continuous Lyapunov equation* is $AX + XA^* + Q = 0$ for an unknown $n \\times n$ matrix $X$.\n2. The *discrete Lyapunov equation* is $AXA^* - X + Q = 0$ for an unknown $n \\times n$ matrix $X$.\n\n## Remarks\n\nThe continuous equation is the [[Sylvester equation|M9SW4NVP2H]] $AX + XA^* = -Q$.\nMany texts in control write it as $A^* P + PA + Q = 0$ instead, which is the same equation with $A^*$ in place of $A$; for real $A$ this reads $A^T P + PA + Q = 0$.\nIf every eigenvalue of $A$ has negative real part, the continuous equation has the unique solution $X = \\int_0^\\infty e^{At} Q e^{A^* t} \\, dt$, and $X$ is positive definite when $Q$ is.\nIf every eigenvalue of $A$ has absolute value less than $1$, the discrete equation has the unique solution $X = \\sum_{k=0}^{\\infty} A^k Q (A^*)^k$.\nExample: for $n = 1$, $A = -1$ and $Q = 2$, the continuous equation $-2X + 2 = 0$ gives $X = 1$.\nWithout the eigenvalue condition a solution may fail to exist: for $n = 1$, $A = 0$ and $Q = 1$, the continuous equation reads $1 = 0$.\n",
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