Submitted Paper

Inserted: 8 dec 2025
Last Updated: 8 dec 2025

Year: 2025

Abstract:

We investigate joint spectral characteristics of a family of matrices $\mathcal{F}$, associated with products in the semigroup generated by $\mathcal{F}$. In the literature, extremal measures such as the well-known joint spectral radius and the lower spectral radius have been extensively studied. However, these measures fail to capture the typical growth rate of matrix products, focusing instead on the worst and best-case scenarios. Nevertheless, when examining, for instance, a switching dynamical system, a probabilistic rate of growth, which characterizes typical trajectories, emerges as a highly intriguing and significant measure. In this article, we present, to the best of our knowledge, the first rigorous analysis of the random spectral radius. This joint spectral characteristic is computed on the set of length-$n$ products from a semigroup by random sampling according to a given probability measure. We establish asymptotic results-including a Law of Large Numbers and a Central Limit Theorem-for cases where the matrices are either diagonal (equivalently, commuting), upper- or lower-triangular, or small perturbations of diagonal matrices. Subsequently, we provide numerical evidence that the random spectral radius of arbitrary (that is structurally unconstrained) families of matrices exhibits asymptotic behavior similar to that of diagonal or nearly diagonal matrix families.