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  "body": "\n## Summary\n\nA Hurwitz-stable matrix is a [[square matrix|0JXZKX526R]] whose [[eigenvalues|0BQAA2BK3Q]] all have negative real part.\nThese are exactly the matrices $A$ for which every solution of $x' = A x$ tends to $0$.\nThrough linearisation they also decide the [[exponential stability|VY180V8QY2]] of equilibria of nonlinear systems.\nThey are also called stable matrices or Hurwitz matrices.\nSee the [[definition|R3VWS51XY5]].\n\n## Explanations\n\n### L1 human\n\nEach eigenvalue $\\lambda$ of $A$ contributes terms like $e^{\\lambda t}$ to the solutions of $x' = Ax$.\nWhen the real part of $\\lambda$ is negative, such a term dies out, possibly while rotating.\nSo a Hurwitz-stable matrix is one whose linear system always returns to rest.\nFor example, $\\begin{pmatrix} -1 \u0026 2 \\\\ 0 \u0026 -3 \\end{pmatrix}$ is Hurwitz-stable, with eigenvalues $-1$ and $-3$.\n",
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