Tight Bounds for Consensus Systems Convergence

We analyze the asymptotic convergence of all infinite products of matrices taken in a given finite set, by looking only at finite or periodic products. It is known that when the matrices of the set have a common nonincreasing polyhedral norm, all infinite products converge to zero if and only if all infinite periodic products with period smaller than a certain value converge to zero, and bounds exist on that value. We provide a stronger bound holding for both polyhedral norms and polyhedral seminorms. In the latter case, the matrix products do not necessarily converge to 0, but all trajectories of the associated system converge to a common invariant space. We prove our bound to be tight, in the sense that for any polyhedral seminorm, there is a set of matrices such that not all infinite products converge, but every periodic product with period smaller than our bound does converge. Our technique is based on an analysis of the combinatorial structure of the face lattice of the unit ball of the nonincreasing seminorm. The bound we obtain is equal to half the size of the largest antichain in this lattice. Explicitly evaluating this quantity may be challenging in some cases. We therefore link our problem with the Sperner property: the property that, for some graded posets, -- in this case the face lattice of the unit ball -- the size of the largest antichain is equal to the size of the largest rank level. We show that some sets of matrices with invariant polyhedral seminorms lead to posets that do not have that Sperner property. However, this property holds for the polyhedron obtained when treating sets of stochastic matrices, and our bound can then be easily evaluated in that case. In particular, we show that for the dimension of the space $n \geq 8$, our bound is smaller than the previously known bound by a multiplicative factor of $\frac{3}{2 \sqrt{\pi n}}$

On the Finiteness Property for Rational Matrices

We analyze the periodicity of optimal long products of matrices. A set of matrices is said to have the finiteness property if the maximal rate of growth of long products of matrices taken from the set can be obtained by a periodic product. It was conjectured a decade ago that all finite sets of real matrices have the finiteness property. This conjecture, known as the ``finiteness conjecture", is now known to be false but no explicit counterexample to the conjecture is available and in particular it is unclear if a counterexample is possible whose matrices have rational or binary entries. In this paper, we prove that finite sets of nonnegative rational matrices have the finiteness property if and only if \emph{pairs} of \emph{binary} matrices do. We also show that all {pairs} of $2 \times 2$ binary matrices have the finiteness property. These results have direct implications for the stability problem for sets of matrices. Stability is algorithmically decidable for sets of matrices that have the finiteness property and so it follows from our results that if all pairs of binary matrices have the finiteness property then stability is decidable for sets of nonnegative rational matrices. This would be in sharp contrast with the fact that the related problem of boundedness is known to be undecidable for sets of nonnegative rational matrices.Comment: 12 pages, 1 figur

On the complexity of computing the capacity of codes that avoid forbidden difference patterns

We consider questions related to the computation of the capacity of codes that avoid forbidden difference patterns. The maximal number of $n$-bit sequences whose pairwise differences do not contain some given forbidden difference patterns increases exponentially with $n$. The exponent is the capacity of the forbidden patterns, which is given by the logarithm of the joint spectral radius of a set of matrices constructed from the forbidden difference patterns. We provide a new family of bounds that allows for the approximation, in exponential time, of the capacity with arbitrary high degree of accuracy. We also provide a polynomial time algorithm for the problem of determining if the capacity of a set is positive, but we prove that the same problem becomes NP-hard when the sets of forbidden patterns are defined over an extended set of symbols. Finally, we prove the existence of extremal norms for the sets of matrices arising in the capacity computation. This result makes it possible to apply a specific (even though non polynomial) approximation algorithm. We illustrate this fact by computing exactly the capacity of codes that were only known approximately.Comment: 7 pages. Submitted to IEEE Trans. on Information Theor

A Gel'fand-type spectral radius formula and stability of linear constrained switching systems

Using ergodic theory, in this paper we present a Gel'fand-type spectral radius formula which states that the joint spectral radius is equal to the generalized spectral radius for a matrix multiplicative semigroup \bS^+ restricted to a subset that need not carry the algebraic structure of \bS^+. This generalizes the Berger-Wang formula. Using it as a tool, we study the absolute exponential stability of a linear switched system driven by a compact subshift of the one-sided Markov shift associated to \bS.Comment: 16 pages; to appear in Linear Algebra and its Application

Efficient algorithms for deciding the type of growth of products of integer matrices

For a given finite set $\Sigma$ of matrices with nonnegative integer entries we study the growth of $$\max_t(\Sigma) = \max\{\|A_{1}... A_{t}\|: A_i \in \Sigma\}.$$ We show how to determine in polynomial time whether the growth with $t$ is bounded, polynomial, or exponential, and we characterize precisely all possible behaviors.Comment: 20 pages, 4 figures, submitted to LA

Computationally efficient approximations of the joint spectral radius

The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set. This quantity appears in a number of application contexts but is notoriously difficult to compute and to approximate. We introduce in this paper a procedure for approximating the joint spectral radius of a finite set of matrices with arbitrary high accuracy. Our approximation procedure is polynomial in the size of the matrices once the number of matrices and the desired accuracy are fixed