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  "body": "\n## Definition\n\nLet $A$ be an $n \\times n$ [[square matrix|0JXZKX526R]] with real or complex entries.\nThe matrix $A$ is *Hurwitz-stable* if every [[eigenvalue|0BQAA2BK3Q]] $\\lambda$ of $A$ satisfies $\\operatorname{Re} \\lambda \u003c 0$.\n\n## Remarks\n\nEquivalently, $e^{tA} \\to 0$ as $t \\to \\infty$, that is, the origin is asymptotically stable for $x' = Ax$.\nThen there are $C \\ge 1$ and $\\alpha \u003e 0$ with $\\norm{e^{tA}} \\le C e^{-\\alpha t}$ for all $t \\ge 0$.\nA real matrix $A$ is Hurwitz-stable if and only if, for one (equivalently, every) positive definite matrix $Q$, the [[Lyapunov equation|92GVPZH9FT]] $A^T P + P A = -Q$ has a positive definite solution $P$.\n\nExample: $A = \\begin{pmatrix} -1 \u0026 2 \\\\ 0 \u0026 -3 \\end{pmatrix}$.\nIt is triangular, so its eigenvalues are its diagonal entries $-1$ and $-3$, both with negative real part.\n\nNon-example: $A = \\begin{pmatrix} 0 \u0026 1 \\\\ -1 \u0026 0 \\end{pmatrix}$.\nIts eigenvalues are $\\pm i$, with real part $0$, so $A$ is not Hurwitz-stable; the solutions of $x' = Ax$ are rotations that do not decay.\n\nFor a real matrix, the condition can be tested on the coefficients of the characteristic polynomial by the Routh–Hurwitz criterion.\n",
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