Theory Of Nonnegative Matrices Research Articles

Conditions of stability and guaranteed convergence rate for linear time-varying discrete-time delay systems

This work introduces novel results on the stability of linear time-varying discrete-time delay systems, addressing both internal stability and input-to-state stability. We start from the special case of positive delay systems and provide conditions of global exponential stability, both delay-dependent and delay-independent. Moreover, we formulate an explicit bound on the exponential decay rate of the state evolution, as a function of the maximum delay. We then show that the aforementioned stability conditions also prove input-to-state stability. The stability conditions consist of simple linear inequalities, albeit infinite-many due to the time-varying nature of the system. To overcome the computational burden, we also introduce a simple relaxation that allows to check the stability with a finite test, at the expense of conservatism. Finally, leveraging comparison systems and results from the theory of non-negative matrices, we generalize all the aforementioned results to systems with no sign constraint, for which we still provide stability conditions by means of linear inequalities and guaranteed convergence rate. Some consequences and a comparative example are provided for the well investigated special case of systems with constant matrices and time-varying delays.

Stability analysis of switched ARX models and application to learning with guarantees

The main subject of this work is the stability analysis of Switched Auto-Regressive models with eXogenous inputs (SARX), which constitute a reference class for switched and hybrid system identification. The work introduces novel conditions for the arbitrary switching stability of multiple-input multiple-output SARX models which exploit the peculiar structure of their state-space realization. The analysis relies on the properties of block companion matrices, and partly leverages results from the theory of non-negative matrices, without nevertheless asking for an input–output positive behavior of the model. The novel stability conditions have a simple formulation in terms of linear co-positive common Lyapunov functions, and come at a remarkably low computational cost, being solvable by Linear Programming. The low computational burden is particularly attractive in an identification context, as it allows to efficiently constrain learning procedures in order to obtain SARX models with stability guarantees. The latter is itself a contribution of the work, as it fills a gap in the literature on the estimation of SARX models. The results are validated on a particular learning technique based on Regression Trees – a well known machine learning algorithm – which has shown remarkable accuracy in experimental environments.

Matrix Analysis for Continuous-Time Markov Chains

AbstractContinuous-time Markov chains have transition matrices that vary continuously in time. Classical theory of nonnegative matrices, M-matrices and matrix exponentials is used in the literature to study their dynamics, probability distributions and other stochastic properties. For the benefit of Perron-Frobenius cognoscentes, this theory is surveyed and further adapted to study continuous-time Markov chains on finite state spaces.

Upper bounds on scrambling index for non-primitive digraphs

The notion of the scrambling index is a fundamental invariant in graph theory and in the theory of non-negative matrices and their applications. Namely, a scrambling index of a primitive directed graph G is the smallest positive integer such that for any pair of vertices u,v of G there exists a vertex w of G such that there are directed walks of length k from u to w and from v to w. In this paper, we generalize the definition to arbitrary directed graphs. We describe constructively the class of graphs with non-zero scrambling index and generalize the Akelbek–Kirkland bounds for the scrambling index to arbitrary directed graphs. Also, the directed graphs with extremal scrambling index are characterized.

Does averaging overestimate or underestimate population growth? It depends

When a matrix population model is nonautonomous, i.e., when it represents a set of single-time-step ("annual") PPMs, L(t), t = 0, 1, …, T – 1, each corresponding to a fixed life cycle graph, then each of the annual matrices generates its own set of model results to characterize the population. In particular, λ1(t), the asymptotic growth rate, varies with t and may result in alternating predictions of population viability. A logical way to characterize the population over the total period of observations is to average the given set of T PPMs, and I have proved the correct mode of averaging to be the pattern-geometric average. It means finding a matrix, G, such that its Tth power equals the product of T annual matrices (in the chronological order), while its pattern does correspond to the given life cycle graph. In practical cases however, the mathematical problem of pattern-geometric average has no exact solution for a fundamental mathematical reason. Nevertheless, the approximate solutions have revealed a fair precision of approximation in recent case studies of alpine short-lived perennials (Eritrichium caucasicum and Androsace albana), resulting in quite certain predictions of population viability by means of λ1(G), the dominant eigenvalue of the average matrix.An alternative approach to estimate the viability leades to the concept of stochastic environment (represented by the given PPMs to be chosen at random) and the ensuing stochastic growth rate, λS. In the case studies, the λSs have been estimated by a direct method of Monte Carlo simulations, all the E. caucasicum estimates having unambiguously been less than λ1(G), whereas those for A. albana being certainly greater than λ1(G). There should be a general mathematical reason for this, too, thus a challenge to the theory of nonnegative matrices.

Conjecture O holds for the odd symplectic Grassmannian

Let $\mathrm{IG}(k, 2n+1)$ be the odd-symplectic Grassmannian. Property $\mathcal{O}$, introduced by Galkin, Golyshev and Iritani for arbitrary complex, Fano manifolds $X$, is a statement about the eigenvalues of the linear operator obtained by the quantum multiplication by the anticanonical class of $X$. We prove that property $\mathcal{O}$ holds in the case when $X= \mathrm{IG}(k, 2n+1)$ is an odd-symplectic Grassmannian. The proof uses the combinatorics of the recently found quantum Chevalley formula for $\mathrm{IG}(k, 2n+1)$, together with the Perron-Frobenius theory of nonnegative matrices.

Global dynamics for a multi-group alcoholism model with public health education and alcoholism age.

A new multi-group alcoholism model with public health education and alcoholism age is considered. The basic reproduction number R₀ is defined and mathematical analyses show that dynamics of model are determined by the basic reproduction number. The alcohol-free equilibrium P₀ of the model is globally asymptotically stable if R₀≤1 while the alcohol-present equilibrium P* of the model exists uniquely and is globally asymptotically stable if R₀>1. The Lyapunov functionals for the globally asymptotically stable of the multi-group model are constructed by using the theory of non-negative matrices and a graph-theoretic approach. Meanwhile, the combined effects of the public health education and the alcoholism age on alcoholism dynamics are displayed. Our main results show that strengthening public health education and decreasing the age of the alcoholism are very helpful for the control of alcoholism.

Primitive tensors and directed hypergraphs

Primitivity is an important concept in the spectral theory of nonnegative matrices and tensors. It is well-known that an irreducible matrix is primitive if and only if the greatest common divisor of all the cycles in the associated directed graph is equal to 1. The main aim of this paper is to establish a similar result, i.e., we show that a nonnegative tensor is primitive if and only if the greatest common divisor of all the cycles in the associated directed hypergraph is equal to 1.

Analysis and Minimization of Interaction in Decentralized Control Systems With Application to Roll-to-Roll Manufacturing

Analysis and minimization of interaction in multivariable systems employing decentralized controllers with application to roll-to-roll manufacturing systems are considered in this paper. A new interaction metric based on the Perron-Frobenius theory of nonnegative matrices is presented. This new metric may be used to quantify interaction in a large-scale interconnected system, establish constraints on closed-loop system stability, and provide a systematic design procedure for constructing decentralized pre-filters, which minimize interaction. The new interaction metric is applied to a roll-to-roll (R2R) manufacturing system, which utilizes decentralized control systems. R2R manufacturing is a continuous process in which flexible materials are transported on rollers through processing machinery where operations, such as printing, coating, lamination, etc., are performed to obtain finished products. Based on the Perron root interaction metric (PRIM), a comprehensive experimental study to analyze and minimize interaction on a large experimental R2R platform is presented. A representative sample of experimental results which demonstrate the applicability of PRIM is presented and discussed.

Formal Power Series and the Spectra of Nonnegative Real Matrices

One of the research interests of Trevor West was the Perron-Frobenius theory of nonnegative matrices and its extensions to infinite dimensions. We present some connections between the characteristic polynomials of nonnegative matrices A and the sign patterns of the Taylor expansions about t = 0 of fractional powers of det (I tA).

FORMAL POWER SERIES AND THE SPECTRA OF NONNEGATIVE REAL MATRICES

One of the research interests of Trevor West was the Perron-Frobenius theory of nonnegative matrices and its extensions to infinite dimensions. We present some connections between the characteristic polynomials of nonnegative matrices A and the sign patterns of the Taylor expansions about t = 0 of fractional powers of det(I-tA).

A new approach for global controllability of higher order Boolean control network

Using the semi-tensor product, the global control problem of the higher order Boolean control network with avoiding set is considered. First, the number of different control sequences that drive the higher order Boolean control network from the initial state to the destination state while avoiding undesirable set is provided. Then, the definition of global controllability of higher order Boolean control network is given. Third, by using the classical theory of nonnegative matrices, the necessary and sufficient condition for the global controllability of higher order Boolean control network with avoiding set is presented. Furthermore, a sufficient condition for k fixed-time global controllability of the system is also obtained. At last, an example is given to illustrate the main results.

On the Perron–Frobenius theory for complex matrices

We extend here the Perron–Frobenius theory of nonnegative matrices to certain complex matrices. Following the generalization of the Perron–Frobenius theory to matrices that have some negative entries, given by Noutsos [14], we introduce here two types of extensions of the Perron–Frobenius theory to complex matrices. We present and prove here some sufficient conditions and some necessary and sufficient conditions for a complex matrix to have a Perron–Frobenius eigenpair. We apply this theory by introducing Perron–Frobenius splittings, as well as complex Perron–Frobenius splittings, for the solution of complex linear systems Ax=b, by classical iterative methods. Perron–Frobenius splittings constitute an extension of the well-known regular splittings, weak regular splittings and nonnegative splittings. Convergence and comparison properties are also given and proved.

A Maximum Principle for the Stability Analysis of Positive Bilinear Control Systems with Applications to Positive Linear Switched Systems

We consider a continuous-time bilinear control system (BCS) with Metzler matrices. Each entry in the transition matrix of such a system is nonnegative, making the positive orthant an invariant set of the dynamics. Motivated by the stability analysis of positive linear switched systems (PLSSs), we define a control as optimal if, for a fixed final time, it maximizes the spectral radius of the transition matrix. Our main result is a first-order necessary condition for optimality in the form of a maximum principle (MP). The proof of this MP combines the standard needle variation with a basic result from the Perron--Frobenius theory of nonnegative matrices. We describe several applications of this MP to the stability analysis of PLSSs under arbitrary switching.

A spectral approach to consecutive pattern-avoiding permutations

We consider the problem of enumerating permutations in the symmetric group on n elements which avoid a given set of consecutive patterns S, and in particular computing asymptotics as n tends to infinity. We develop a general method which solves this enumeration problem using the spectral theory of integral operators onL2([0,1]m), where the patterns in S have length m+1.Kre\u{\i}n and Rutman’s generalization of the Perron–Frobenius theory of non-negative matrices plays a central role. Our methods give detailed asymptotic expansions and allow for explicit computation of leading terms in many cases.As a corollary to our results,we settle a conjecture of Warlimont on asymptotics for the number of permutations avoiding a consecutive pattern.

The Algebraic Degree of Phase-Type Distributions

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

The Algebraic Degree of Phase-Type Distributions

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

Conditions for some balances of economic system: An input–output analysis using the spectral theory of nonnegative matrices

Twelve kinds of special semipositive matrices and their basic characters are presented. Employing these matrices and the previous results in Zeng (2008), we research the conditions for the balances between final output values and values-added, and between input multipliers and output multipliers in an economy. A necessary and sufficient condition that (i) there exists a unique vector of output adjustment coefficients such that (a) all sectoral final output values equal their respective sectoral values-added in the new output system, or (b) all sectoral input multipliers redefined by the new output system equal their respective sectoral output multipliers; or (ii) there exists a unique vector of price adjustment coefficients such that (a) all sectoral values-added equal their respective sectoral final output values in the new price system, or (b) all sectoral output multipliers redefined by the new price system equal their respective sectoral input multipliers; is the irreducibility of the matrix of intermediate output (or input) coefficients. A necessary and sufficient condition that (i) there is no vector of output adjustment coefficients which enables all sectoral final output values (or input multipliers) to equal their respective sectoral values-added (or output multipliers), or (ii) there is no vector of price adjustment coefficients which enables all sectoral values-added (or output multipliers) to equal their respective sectoral final output values (or input multipliers); is that the matrix of intermediate output (or input) coefficients has at least one non-final (or non-initial) class. These class relations and their equivalent conditions are summarized. The elaborate examples verify the main conclusions thoroughly.

Global threshold property in an epidemic model for disease with latency spreading in a heterogeneous host population

In this paper, a general mathematical model is proposed with detailed justifications to describe the spread of a disease with latency in a heterogeneous host population which includes many existing ones as special cases. For a simpler version that assumes an identical natural death rate for all groups, and with a gamma distribution for the latency, the model is shown to demonstrate the global threshold dynamics in terms of the basic reproduction number R 0 of the model: if R 0 ≤ 1 , the disease-free equilibrium is globally asymptotically stable in the positive orthant, whereas if R 0 > 1 , a unique endemic equilibrium exists and is globally asymptotically stable in the interior of the positive orthant. The proofs of the main results make use of the theory of non-negative matrices, persistence theory in dynamical systems, Lyapunov functions and a subtle grouping technique in estimating the derivatives of Lyapunov functions guided by graph theory, which was recently developed and applied by several authors to some relateted epidemic models.

The Rate of Rényi Entropy for Irreducible Markov Chains

In this paper, we obtain the Rényi entropy rate for irreducible-aperiodic Markov chains with countable state space, using the theory of countable nonnegative matrices. We also obtain the bound for the rate of Rényi entropy of an irreducible Markov chain. Finally, we show that the bound for the Rényi entropy rate is the Shannon entropy rate.