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  "body": "\n## Statement\n\nLet $p(s) = a_0 s^n + a_1 s^{n-1} + \\cdots + a_{n-1} s + a_n$ be a polynomial of degree $n \\ge 1$ with real coefficients and $a_0 \u003e 0$.\nPut $a_k = 0$ for $k \u003c 0$ and for $k \u003e n$.\nLet $H$ be the $n \\times n$ Hurwitz matrix with entries $H_{ij} = a_{2j - i}$ for $1 \\le i, j \\le n$, and let $\\Delta_k$ be its leading principal [[minor|0BT9BYWVRY]] of order $k$.\nThen every [[root|019MA1VTR2]] of $p$ has negative real part if and only if $\\Delta_k \u003e 0$ for $k = 1, \\ldots, n$.\n\n## Remarks\n\nThe rows of $H$ are $(a_1, a_3, a_5, \\ldots)$, $(a_0, a_2, a_4, \\ldots)$, $(0, a_1, a_3, \\ldots)$, $(0, a_0, a_2, \\ldots)$, and so on.\nOne proof counts the roots of $p$ in the right half-plane by following the argument of $p(i\\omega)$ as $\\omega$ runs over $\\R$.\nRouth's array computes the same information by a Euclidean algorithm on the even and odd parts of $p$.\n\nFor $n = 2$ and $p(s) = s^2 + a_1 s + a_2$, the conditions are $a_1 \u003e 0$ and $a_1 a_2 \u003e 0$, that is, $a_1 \u003e 0$ and $a_2 \u003e 0$.\nFor $n = 3$, they are $a_1 \u003e 0$, $a_1 a_2 - a_0 a_3 \u003e 0$ and $a_3 \u003e 0$, since $\\Delta_3 = a_3 \\Delta_2$.\nPositive coefficients are necessary but not sufficient when $n \\ge 3$.\nFor $p(s) = s^3 + s^2 + s + 1 = (s + 1)(s^2 + 1)$, all coefficients are positive, but $\\Delta_2 = 1 - 1 = 0$, and indeed $\\pm i$ are roots.\n\n## Explanations\n\n### L1 human\n\nTo know whether a linear system returns to rest, one must know whether all roots of its characteristic polynomial lie in the left half-plane.\nThe criterion answers this from the coefficients alone, by checking that some determinants are positive.\nNo root has to be computed.\n\n### L2 human\n\nApplied to the [[characteristic polynomial|0VSSPAFM2C]] of a real matrix $A$, the criterion tests whether $A$ is [[Hurwitz-stable|9A3TBBHRFA]].\nWith the [[linearisation|09ZWYRQ41J]] of a nonlinear system, it gives a test for the stability of an equilibrium that uses only arithmetic on the entries of the Jacobian matrix.\nIt is the standard tool in control engineering for stability with respect to parameters, since the conditions are explicit inequalities.\nThe analogous test for roots inside the unit disc, used for discrete-time systems, is the Schur–Cohn or Jury test.\n",
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  "name": "Routh–Hurwitz stability criterion",
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