Probabilistic Analysis of the Random Spectral Radius for a Matrix Family

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 study the random spectral radius, defined as the spectral radius of a length-$n$ product sampled at random from the semigroup according to a given probability measure. We establish asymptotic results, namely a Law of Large Numbers and a Central Limit Theorem, for diagonal (equivalently, commuting), upper- or lower-triangular, and small perturbations of diagonal matrices. Beyond recovering the correct scaling, we obtain exact closed-form expressions for the limiting value and variance, which may be useful in numerical applications requiring precise constants. In the coalescence regime, where the leading eigenvalues merge, the limiting distribution is non-Gaussian: it is given by the maximum of a correlated Gaussian vector with explicit covariance structure. This phenomenon governs phase transitions between distinct growth regimes in switching systems.