= Joint spectral radius
{title2=$\widehat\rho(\mathcal A)=\lim_{p\to\infty}\max_{A_i\in\mathcal A}\|A_p\cdots A_1\|^{1/p}$}
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The <joint spectral radius> measures the worst exponential growth under arbitrary products from a finite family of <matrices>. Submultiplicativity gives its existence and invariance under changes of the chosen norm. Every fixed-length product $B$ satisfies $\rho(B)^{1/p}\le\widehat\rho(\mathcal A)$, while the maximum norm of all length-$p$ products to the power $1/p$ is an <upper bound>. A repeated product gives the first inequality; grouping an arbitrary long word into blocks proves the second. The <spectral radius> of each individual <matrix> does not control growth of all switched products.
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