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  "body": "\n## Summary\n\nA Schur stable polynomial is a polynomial whose [[roots|019MA1VTR2]] all lie strictly inside the unit circle of the complex plane.\nIt is the discrete-time partner of a Hurwitz stable polynomial, whose roots lie in the open left half-plane.\nA linear recurrence with constant coefficients has all its solutions tending to zero exactly when its [[characteristic polynomial|0YA7X86K05]] is Schur stable.\nSo the notion is the basic test for stability of discrete-time linear systems and of numerical schemes.\n\n## Explanations\n\n### L1 human\n\nLook at all the complex roots of a polynomial and draw them in the plane.\nThe polynomial is Schur stable when every root lies inside the circle of radius $1$ around the origin, and none lies on the circle.\nFor example, $z^2 - \\tfrac{1}{4}$ has roots $\\pm \\tfrac{1}{2}$, so it is Schur stable.\nThe polynomial $z - 2$ is not, because its root $2$ lies outside the circle.\nPowers $\\lambda^n$ of a root with $\\abs{\\lambda} \u003c 1$ shrink to zero, which is why these polynomials describe systems that settle down.\n",
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