Polynomial-Time Computation of the Joint Spectral Radius for Some Sets of Nonnegative Matrices

Abstract

Abstract. We propose two simple upper bounds for the joint spectral radius of sets of nonnegative matrices. These bounds, the joint column radius and the joint row radius, can be computed in polynomial time as solutions of convex optimization problems. We show that these bounds are within a factor 1/n of the exact value, where n is the size of the matrices. Moreover, for sets of matrices with independent column uncertainties or with independent row uncertainties, the corresponding bounds coincide with the joint spectral radius. In these cases, the joint spectral radius is also given by the largest spectral radius of the matrices in the set. As a by-product of these results, we propose a polynomial-time technique for solving Boolean optimization problems related to the spectral radius. We also describe economics and engineering applications of our results.

Hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of non-negative matrices

“…As was noted in Section 1, one of the most interesting classes of matrices for which the finiteness conjecture holds, both for the generalized and lower spectral radius, is the so-called class of non-negative matrices with independent row uncertainty [1]. In this section, we recall the relevant definition and present a new proof of the corresponding results on finiteness needed to motivate further constructions.…”

“…In particular, this class includes all the sets of self-adjoint matrices. One of the most interesting classes of matrices for which the finiteness conjecture is valid, for both the generalized and the lower spectral radius, is the so-called class of non-negative matrices with independent row uncertainty [1]. Note that in all these cases, the generalized spectral radius coincides with the spectral radius of a single matrix from A or with the spectral radius of the product of a pair of such matrices.…”

“…In particular, in the course of transposing, the sets of matrices with independent row uncertainty turn into the so-called sets of matrices with independent column uncertainty [1].…”

“…Following [1], a set of matrices A ⊂ M(N, M ) is called an independent row uncertainty set (IRU-set) if it consists of all the matrices…”

“…

AbstractRecently Blondel, Nesterov and Protasov proved [1,2] that the finiteness conjecture holds for the generalized and the lower spectral radii of the sets of non-negative matrices with independent row/column uncertainty. We show that this result can be obtained as a simple consequence of the so-called hourglass alternative used in [3], by the author and his companions, to analyze the minimax relations between the spectral radii of matrix products.

…”