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  "body": "\n## Definition\n\nLet $A$ be an $n \\times n$ complex [[square matrix|0JXZKX526R]], and let $\\sigma(A) \\subset \\C$ be its [[spectrum|0XPX5NC67E]], the set of its [[eigenvalues|0BQAA2BK3Q]].\nThe *spectral radius* of $A$ is\n$$\\rho(A) = \\max \\set{\\abs{\\lambda} : \\lambda \\in \\sigma(A)}.$$\n\n## Remarks\n\nExample: $A = \\begin{pmatrix} 0 \u0026 -1 \\\\ 1 \u0026 0 \\end{pmatrix}$ has characteristic polynomial $x^2 + 1$, so $\\sigma(A) = \\set{i, -i}$ and $\\rho(A) = 1$.\n\nThe spectral radius is not a norm on matrices.\nFor $N = \\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$, the only eigenvalue is $0$, so $\\rho(N) = 0$ although $N \\ne 0$.\n\nFor every [[operator norm|06EKW1AKXF]] $\\norm{\\cdot}$, $\\rho(A) \\le \\norm{A}$, and [[Gelfand's formula|3KX0KQ7V05]] gives $\\rho(A) = \\lim_{k \\to \\infty} \\norm{A^k}^{1/k}$.\nThe powers $A^k$ tend to $0$ if and only if $\\rho(A) \u003c 1$.\nFor a [[normal matrix|0MR0BYF0K2]], $\\rho(A)$ equals the operator norm induced by the Euclidean norm.\nThe same definition applies to a linear map of a finite-dimensional complex vector space.\n\n## Explanations\n\n### L1 human\n\nThe spectral radius is the size of the largest eigenvalue.\nIt says how fast repeated application of the matrix makes vectors grow or shrink in the long run.\n",
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