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  "body": "\n## Summary\n\nThe spectral radius of a square complex matrix is the largest absolute value of its [[eigenvalues|0BQAA2BK3Q]].\nIt controls the long-run growth of the powers $A^k$: they tend to $0$ exactly when the spectral radius is less than $1$.\nIt is bounded by every operator norm, and [[Gelfand's formula|3KX0KQ7V05]] recovers it from the norms of the powers.\n\n## Explanations\n\n### L1 human\n\nDraw the eigenvalues of a matrix in the complex plane.\nThe spectral radius is the radius of the smallest disc around $0$ that contains all of them.\nFor $\\begin{pmatrix} 2 \u0026 0 \\\\ 0 \u0026 -3 \\end{pmatrix}$ the eigenvalues are $2$ and $-3$, so the spectral radius is $3$.\nApplying the matrix many times makes vectors grow roughly like $3^k$.\n",
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