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  "body": "\n## Summary\n\nA submultiplicative function is a nonnegative function $f$ on a [[semigroup|0XMEM406JM]] with $f(xy) \\le f(x) f(y)$ for all $x$ and $y$.\nIt is the multiplicative twin of a [[subadditive function|3X9DD9J3CV]]: where $f$ is positive, $\\log f$ is subadditive.\nThe [[operator norm|06EKW1AKXF]] on square matrices is submultiplicative.\nSo is the Frobenius norm, a [[matrix norm|54T6Y6DSY4]] that is not an operator norm.\nWith Fekete's subadditive lemma this gives the existence of limits such as $\\lim_n \\norm{T^n}^{1/n}$.\n\n## Explanations\n\n### L1 human\n\nA submultiplicative quantity never grows faster under composition than the product of its values.\nThink of a machine that stretches vectors by at most a factor $a$ followed by one that stretches by at most $b$: together they stretch by at most $ab$.\nFor example, the sequence $a_n = \\norm{A^n}$ of norms of powers of a matrix $A$ satisfies $a_{m+n} \\le a_m a_n$.\nSo the growth of $\\norm{A^n}$ is controlled by a single number, the limit of $a_n^{1/n}$.\n",
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