Question about the joint spectral radius

Question

Given a bounded set $\mathcal A\subset \Bbb R^{n \times n}$, the joint spectral radius is given by

$$\sigma(\mathcal A) = \limsup_{m \to \infty} \left( \,\sup_{A \in \mathcal A^m} \rho(A) \right)$$

where $\rho$ is the normal spectral radius and

$$\mathcal A^m = \{ A_1 A_2 \cdots A_m : A_i \in \mathcal A, i=1,\dots,m\}$$

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I have problems understanding what the $m$ means? For example if I have given $\mathcal A$={$A_1,A_2$} set of 2 matrices is then $m=2$ and $\mathcal A^2 = \{ A_1 A_2 \}$ or can $m$ be greater? What is the bound of $m$? If for example $m=5$ is then $\mathcal A^5 = \{ A_1 A_2^2 A_1^2\}$?

Answer 1

$m$ can be any natural number, in case of $\mathcal A =\{A_1, A_2\}$ we have

• $\mathcal A^2 = \{A_1^2, A_1A_2, A_2A_1, A_2^2\}$
• $\mathcal A^3 = \{A_1^3, A_1^2A_2, A_1A_2A_1, A_2A_1^2, A_1A_2^2, A_2A_1A_2, A_2^2A_1, A_2^3\}$

and so on. $\mathcal A^m$ consists of all $m$-fold products of matrices from $\mathcal A$, for example, $A_1A_2^2A_1^2 \in \mathcal A^5$, but $\mathcal A^5$ has of course more elements.