Infinite Nonnegative Matrices Research Articles

Containment control for heterogeneous multi-agent systems with asynchronous updates

This paper is concerned with the asynchronous containment control problem for heterogeneous multi-agent systems with time-varying delays. Here, asynchrony means that each agent updates the state information by its own clock that is independent of the other agents’ update times, and the update intervals of each agent are not necessarily equispaced. It is assumed that the leaders are stationary, and the followers can be classified into two generic categories: the followers with the first-order dynamics and the followers with second-order dynamics. For different kinds of followers, two distributed containment control protocols are presented to guarantee that all the followers can asymptotically converge into the convex hull formed by the leaders. The properties of the product of infinite nonnegative matrices are explored to arrive at that the asynchronous containment control can be achieved under the proposed protocols if and only if the communication topology among the agents contains a directed spanning forest rooted at the leaders. At last, simulation results are given to demonstrate the validity of the theoretical findings.

Inequalities on the spectral radius and the operator norm of Hadamard products of positive operators on sequence spaces

Relatively recently, K.M.R. Audenaert (2010), R.A. Horn and F. Zhang (2010), Z. Huang (2011), A.R. Schep (2011), A. Peperko (2012), D. Chen and Y. Zhang (2015) have proved inequalities on the spectral radius and the operator norm of Hadamard products and ordinary matrix products of finite and infinite non-negative matrices that define operators on sequence spaces. In the current paper we extend and refine several of these results and also prove some analogues for the numerical radius. Some inequalities seem to be new even in the case of $n\times n$ non-negative matrices.

On asymptotic properties of equilibrium measures corresponding to finite submatrices of infinite nonnegative matrices

There is a canonical way to assign a translation invariant Markov measure to every finite irreducible nonnegative matrix. This measure is defined on a sequence space, translation invariant, and satisfies a variational principle (due to the last property it is said to be equilibrium). The same can be done for some infinite nonnegative matrices. If A is such a matrix, one can consider an increasing sequence of its finite irreducible submatrices A n that tends to A in a natural sense, and ask if the equilibrium measure μ An assigned to A n converges to the equilibrium measure μ A assigned to A. For two classes of matrices A, the answer does not depend on {A n }: for one of these classes, μ An ? μ A , for the other one, μ An converges to the zero measure. We describe, in geometric terms, a third class located `between' the above two for which the situation is also intermediate: for some sequences {A n } the asymptotic behavior of μ An is as in the first class, while for some other sequences the behavior is as in the second one.

Bounds on the generalized and the joint spectral radius of Hadamard products of bounded sets of positive operators on sequence spaces

Recently, Audenaert (2010) [2], Horn and Zhang (2010) [15], Huang (2011) [16] and Schep (2011) [22,23] proved inequalities between the spectral radius ρ of Hadamard product (denoted by ∘) of finite and infinite non-negative matrices that define operators on sequence spaces and the spectral radius of their ordinary matrix product. We extend these results to the generalized and the joint spectral radius of bounded sets of such operators. Moreover, we prove new inequalities even in the case of the usual spectral radius of non-negative matrices. In particular, we prove thatρ(A∘B)≤ρ12((A∘A)(B∘B))≤ρ(AB∘AB)14ρ(BA∘BA)14≤ρ(AB)andρ(A∘B)≤ρ12(AB∘BA)≤ρ(AB∘AB)14ρ(BA∘BA)14≤ρ(AB).We also obtain related results in max algebra.

On the functional inequality for the spectral radius of compact operators

Elsner et al. [Functional inequalities for spectral radii of nonnegative matrices, Linear Algebra Appl. 129 (1990), pp. 103–130] characterized functions satisfying for all non-negative matrices A 1, … , A n of the same order, where r denotes the spectral radius. We generalize this result to the setting of infinite non-negative matrices that define compact operators on a Banach sequence space.

Bounds on the spectral radius of Hadamard products of positive operators on lp-spaces

Recently, K.M.R. Audenaert (2010), and R.A. Horn and F. Zhang (2010) proved inequalities between the spectral radius of Hadamard products of finite nonnegative matrices and the spectral radius of their ordinary matrix product. We will prove these inequalities in such a way that they extend to infinite nonnegative matrices A and B that define bounded operators on the classical sequence spaces lp.

On the entropy of context-free languages

The information theoretical concept of the entropy (channel capacity) of context-free languages and its relation to the structure generating function is investigated in the first part of this paper. The achieved results are applied to the family of pseudolinear grammars. In the second part, relations between context-free grammars, infinite labelled digraphs and infinite nonnegative matrices are exhibited. Theorems on the convergence parameter of infinite matrices are proved and applied to the evaluation of the entropy of certain context-free languages. Finally, a stochastic process is associated with any context-free language generated by a deterministic labelled digraph, such that the stochastic process is equivalent to the language in the sense that both have the same entropy.

Finite approximations to infinite non-negative matrices, II: refinements and applications

In a recent paper (1), the author has discussed how certain ergodic properties of a doubly infinite non-negative matrix T = {tij} are related to those of its finite (n × n) corner truncations, (n)T = {(n)tij}.