Rank-one Characterization of Joint Spectral Radius of Finite Matrix Family
This provides theoretical tools for computing joint spectral radius, relevant to control theory and dynamical systems, but the results are incremental and limited to specific matrix classes.
The paper characterizes the joint/generalized spectral radius of finite matrix families via rank-one approximations using singular value decomposition, deriving formulas for families with at most one high-rank matrix and obtaining new characterizations for general matrices, with numerical examples.
In this paper we study the joint/generalized spectral radius of a finite set of matrices in terms of its rank-one approximation by singular value decomposition. In the first part of the paper, we show that any finite set of matrices with at most one element's rank being greater than one satisfies the finiteness property under the framework of (invariant) extremal norm. Formula for the computation of joint/generalized spectral radius for this class of matrix family is derived. Based on that, in the second part, we further study the joint/generalized spectral radius of finite sets of general matrices through constructing rank-one approximations in terms of singular value decomposition, and some new characterizations of joint/generalized spectral radius are obtained. Several benchmark examples from applications as well as corresponding numerical computations are provided to illustrate the approach.