Rank-one Characterization of Joint Spectral Radius of Finite Matrix Family

Abstract

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.

arXiv commentSubmitted for review on April 23, 2011; Final version is updated on Jan. 4, 2013 and Dec. 29, 2016

References in corpus (4)

• Approximation of the joint spectral radius using sum of squares
• Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets
• Rank one matrices do not contribute to the failure of the finiteness property
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Cited by in corpus (3)

• Hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of non-negative matrices
• Lyapunov Exponent of Rank One Matrices: Ergodic Formula and Inapproximability of the Optimal Distribution
• Optimal switching sequence for switched linear systems