Joint spectral radius ($\hat{\rho }(\mathcal{A})={\lim }_{p\to \infty }{\max }_{A_i\in \mathcal{A}}\Vert A_p\cdots A_1{\Vert }^{1/p}$)

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 $\rho (B{)}^{1/p}\le \hat{\rho }(\mathcal{A})$, 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.