Irreducible Matrices Research Articles

Distributed consensus-based estimation of the leading eigenvalue of a non-negative irreducible matrix

This paper presents an algorithm to solve the problem of estimating the largest eigenvalue and its corresponding eigenvector for irreducible matrices in a distributed manner. The proposed algorithm utilizes a network of computational nodes that interact with each other, forming a strongly connected digraph where each node handles one row of the matrix, without the need for centralized storage or knowledge of the entire matrix. Each node possesses a solution space, and the intersection of all these solution spaces contains the leading eigenvector of the matrix. Initially, each node selects a random vector from its solution space, and then, while interacting with its neighbors, updates the vector at each step by solving a quadratically constrained linear program (QCLP). The updates are done so that the nodes reach a consensus on the leading eigenvector of the matrix. The numerical outcomes demonstrate the effectiveness of our proposed method.

Non-invertible planar self-affine sets

Abstract We compare the dimension of a non-invertible self-affine set to the dimension of the respective invertible self-affine set. In particular, for generic planar self-affine sets, we show that the dimensions coincide when they are large and differ when they are small. Our study relies on thermodynamic formalism where, for dominated and irreducible matrices, we completely characterise the behaviour of the pressures.

4 × 4 Irreducible sign pattern matrices that require four distinct eigenvalues

A sign pattern matrix is a matrix whose entries are from the set {+,−,0}. For a sign pattern matrix A, the qualitative class of A, denoted Q(A), is the set of all real matrices whose entries have signs given by the corresponding entries of A. An n×n sign pattern matrix A requires all distinct eigenvalues if every real matrix in Q(A) has n distinct eigenvalues. Li and Harris (2002) [13] characterized the 2×2 and 3×3 irreducible sign pattern matrices that require all distinct eigenvalues, and established some useful general results on n×n sign patterns that require all distinct eigenvalues. In this paper, we characterize 4×4 irreducible sign patterns that require four distinct eigenvalues. This is done by characterizing 4×4 irreducible sign patterns that require four distinct real eigenvalues, that require four distinct nonreal real eigenvalues, or that require two distinct real eigenvalues and a pair of conjugate nonreal eigenvalues. The last case turns out to be much more involved. Some interesting open problems are presented. Three important tools that are used in the paper are the following: the discriminant of a polynomial; the fact that if a square sign pattern matrix A requires all distinct eigenvalues then A requires a fixed number of real eigenvalues; and the known result that if A is a “k-cycle” sign pattern then for each B∈Q(A), the k nonzero eigenvalues of B are evenly distributed on a circle in the complex plane centered at the origin.

Study on the Similarity between Nonnegative Irreducible Matrices and Positive Matrices

The main objective of this study is to explore the problem of similarity between nonnegative irreducible matrices and positive matrices. By leveraging established lemmas and drawing insights from pertinent literature, this article provides a rigorous proof: When is greater than 8, for any nonnegative irreducible matrix , if it contains zero elements and possesses mutually distinct diagonal elements, then is similar to a positive matrix.

Perron-Frobenius type theorem for nonnegative tubal matrices in the sense of t-product

This paper is focused on the study of Perron-Frobenius (PF) type theorem for nonnegative tubal matrices, where a tubal matrix is a matrix with every element being a tubal scalar (a vector in the usual sense). Such a matrix is often recognized as a third-order tensor. The product between tubal scalars, tubal vectors, and tubal matrices can be done by the powerful t-product. In this paper, we define nonnegative/positive/strongly positive tubal scalars/vectors/matrices, and establish several properties that are analogous to their matrix counterparts. In particular, we introduce irreducible tubal matrices and provide two equivalent characterizations. Then, the celebrated PF theorem is established on the nonnegative irreducible tubal matrices. We show that some conclusions of the PF theorem for nonnegative irreducible matrices, such as the existence of a positive eigenvector, can be generalized to the tubal matrix setting, while some are not. For those conclusions that can not be extended, weaker conclusions are proved. We also show that, if the nonnegative irreducible tubal matrix contains a strongly positive tubal scalar, then most conclusions of the matrix PF theorem hold.

Reversible random walks on dynamic graphs

AbstractThis paper discusses random walks on edge‐changing dynamic graphs. We prove general and improved bounds for mixing, hitting, and cover times for a random walk according to a sequence of irreducible and reversible transition matrices with the time‐invariant stationary distribution. An interesting consequence is the tight bounds of the lazy Metropolis walk on any dynamic connected graph. We also prove bounds for multiple random walks on dynamic graphs. Our results extend previous upper bounds for simple random walks on dynamic graphs and give improved and tight upper bounds in several cases. Our results reinforce the observation that time‐inhomogeneous Markov chains with an invariant stationary distribution behave almost identically to a time‐homogeneous chain.

Modified Kleene Star Algorithm Using Max-Plus Algebra and Its Application in the Railroad Scheduling Graphical User Interface

In max-plus algebra, some algorithms for determining the eigenvector of irreducible matrices are the power algorithm and the Kleene star algorithm. In this research, a modified Kleene star algorithm will be discussed to compensate for the disadvantages of the Kleene star algorithm. The Kleene star algorithm’s time complexity is O(n(n!)), and the new Kleene star algorithm’s time complexity is O(n4), while the power algorithm’s time complexity cannot be calculated. This research also applies max-plus algebra in a railroad network scheduling problem, constructing a graphical user interface to perform schedule calculations quickly and easily.

Two-step Noda iteration for irreducible nonnegative matrices

ABSTRACT In this paper, we present a two-step Noda iteration for computing the Perron root and Perron vector of an irreducible nonnegative matrix by successively solving two linear systems with the same coefficient matrix. For every positive initial vector, the two-step Noda iteration always converges and has a cubic asymptotic convergence rate. As an application, the two-step Noda iteration is applied to compute the smallest eigenpair of irreducible nonsingular M-matrices. Numerical examples are provided for testing the proposed method.

Extremal inverse eigenvalue problem for irreducible acyclic matrices

In this paper, we study the inverse eigenvalue problem of constructing symmetric matrices whose graph is a tree, i.e. of constructing irreducible acyclic matrices from given eigendata consisting of the smallest and largest eigenvalues of their leading principal submatrices. We solve the problem completely for the case when the given eigenvalues are distinct and the subgraph of the tree induced by remains connected for . The result generalizes the results known for matrices described by some specific trees. The proofs of the main results are constructive and so provide an algorithm for computing the actual entries of the required matrix. Numerical solutions of the problem are then obtained with SCILAB by feeding the eigendata and plugging in the adjacency matrix as inputs. We also make an attempt to find the number of solutions for several special families of unlabelled trees.

On the Perron root and eigenvectors associated with a subshift of finite type

In this paper, we describe the relationship between the Perron root and eigenvectors of an irreducible subshift of finite type with the correlation between the forbidden words in the subshift. In particular, we derive an expression for the Perron eigenvectors of the associated adjacency matrix. As an application, we obtain the Perron eigenvectors for irreducible (0,1) matrices which are adjacency matrices for directed graphs. Moreover, we derive an alternate definition of the Parry measure in ergodic theory on an irreducible subshift of finite type.

Self-equilibrium and super-stability of rhombic truncated regular tetrahedral and cubic tensegrities using symmetry-adapted force-density matrix method

Tensegrities consisting of axially loaded strings and bars have various technologically important applications, for which symmetric configurations are preferred as prototypes. In this study, we investigate the self-equilibrated states of rhombic truncated regular tetrahedral and cubic tensegrities by combining group representation theory and force-density matrix method. The force-density matrices of these tensegrities are analytically block-diagonalized using the irreducible representation matrices of tetrahedral and cubic groups. By making the symmetry-adapted force-density matrix satisfy the necessary number of nullity and positive semi-definiteness, we derive the analytical expressions for self-equilibrium and super-stability of rhombic truncated regular tetrahedral and cubic tensegrities. This work helps to design highly symmetric tensegrities for developing biomechanical models, mechanical metamaterials, and flexible robotics.

Population projections from holey matrices: Using prior information to estimate rare transition events

Population projection matrices are a common means for predicting short- and long-term population persistence for rare, threatened and endangered species. Data from such species can suffer from small sample sizes and consequently miss rare demographic events resulting in incomplete or biologically unrealistic life cycle trajectories. Matrices with missing values (zeros; e.g., no observation of seeds transitioning to seedlings) are often patched using prior information from the literature, other populations, time periods, other species, best guess estimates, or are sometimes even ignored. To alleviate this problem, we propose using a multinomial-Dirichlet model for parameterizing transitions and a Gamma for reproduction to patch missing values in these holey matrices. This formally integrates prior information within a Bayesian framework and explicitly includes the weight of the prior information on the posterior distributions. We show using two real data sets that the weight assigned to the prior information mainly influences the dispersion of the posteriors, the inclusion of priors results in irreducible and ergodic matrices, and more biologically realistic inferences can be made on the transition probabilities. Because the priors are explicitly stated, the results are reproducible and can be re-evaluated if alternative priors are available in the future.

A decomposition of the tensor product of matrices

In this paper we decompose (under unitary equivalence) the tensor product A ? A into a direct sum of irreducible matrices, when A is a 3 x 3 matrix.

Inclusion regions and bounds for the eigenvalues of matrices with a known eigenpair

AbstractLet (λ, v) be a known real eigenpair of an n×n real matrix A. In this paper it is shown how to locate the other eigenvalues of A in terms of the components of v. The obtained region is a union of Gershgorin discs of the second type recently introduced by the authors in a previous paper. Two cases are considered depending on whether or not some of the components of v are equal to zero. Upper bounds are obtained, in two different ways, for the largest eigenvalue in absolute value of A other than. Detailed examples are provided. Although nonnegative irreducible matrices are somewhat emphasized, the main results in this paper are valid for any n × n real matrix with n≥3.

Extended Irreducible Nekrasov Matrices as Subclasses of Irreducible H-Matrices

In this paper, we introduce the irreducible $$\alpha $$ -Nekrasov matrices and irreducible $$\alpha $$ -S-Nekrasov matrices as the extended irreducible Nekrasov matrices and we analyze the relationships among the involved matrices and irreducible H-matrices.

Generating irreducible copositive matrices using the stable set problem

In this paper it is considered how graphs can be used to generate copositive matrices, and necessary and sufficient conditions are given for these generated matrices to then be irreducible with respect to the set of positive semidefinite plus nonnegative matrices. This is done through combining the well known copositive formulation of the stable set problem with recent results on necessary and sufficient conditions for a copositive matrix to be irreducible. By applying this result, tens of thousands of new irreducible copositive matrices of order less than or equal to thirteen were found. These matrices were then tested for being extreme copositive matrices, and it was found that in all but three cases this was indeed the case. It is also demonstrated in this paper that these matrices provide difficult instances to test approximations of the copositive cone against.

Consensus and Disagreement of Heterogeneous Belief Systems in Influence Networks

Recently, an opinion dynamics model has been proposed to describe a network of individuals discussing a set of logically interdependent topics. For each individual, the set of topics and the logical interdependencies between the topics (captured by a logic matrix) form a belief system. We investigate the role the logic matrix and its structure plays in determining the final opinions, including existence of the limiting opinions, of a strongly connected network of individuals. We provide a set of results that, given a set of individuals' belief systems, allow a systematic determination of which topics will reach a consensus, and which topics will disagreement in arise. For irreducible logic matrices, each topic reaches a consensus. For reducible logic matrices, which indicates a cascade interdependence relationship, conditions are given on whether a topic will reach a consensus or not. It turns out that heterogeneity among the individuals' logic matrices, including especially differences in the signs of the off-diagonal entries, can be a key determining factor. This paper thus attributes, for the first time, a strong diversity of limiting opinions to heterogeneity of belief systems in influence networks, in addition to the more typical explanation that strong diversity arises from individual stubbornness.

D-Matrix: A Novel Ranking Procedure for Prioritizing Data Items

In this article, we propose a ranking method based on a matrix, called D-matrix, with the special identical diagonal values. This ranking system has five properties: (1) it can provide both biased and bias-free ranking, and except for that, the working matrix can be built in two ways: results merging and results separating for both biased and bias-free matrices. (2) it can perform the webpage ranking with a sparse matrix to generate ratings for pages instead of constructing complicated, irreducible, and stochastic matrices as the Google PageRank matrix does, thereby accelerating the computation speed. (3) this D-matrix has a solution no matter how much data is selected. If there are no comparisons among items, then all the items end up with the same equal ratings. (4) the ranking system has the least effects on data variation. If one item changes, only those connecting to it get different ratings, those without connection retain the same ratings. (5) this D-matrix has a $\ddot {R}$ support matrix with a delicate diagonal value which may or may not appear crucial. These five features are illustrated with five different and comprehensive examples. Besides that, a 2017 game of the National Football League data is tested where the D-matrix generates a reasonable result. Furthermore, we introduce a new approximate non-dominated sorting method based on D-matrix and thereby put forth a new algorithm for solving the multi-objective optimization problems. Experimental results indicate that our algorithm can maintain a better spread of solutions on many standard test functions.

On the spectral theory of positive operators and PDE applications

The strong version of the Krein-Rutman theorem requires that the positive cone of the Banach space has nonempty interior and the compact map is strongly positive, mapping nonzero points in the cone into its interior. In this paper, we first generalize this version of the Krein-Rutman theorem to the case of 'semi-strongly positive' operators; and we prove it in a totally elementary fashion. We then prove the equivalence of semi-strong positivity and irreducibility in a Banach lattice, linking the afore-mentioned result with the Krein-Rutman theorem for irreducible operators. One of the things we emphasize is to use 'upper and lower spectral radii' to characterize, in the fashion of Collatz-Wielandt formula for nonnegative irreducible matrices, the principal eigenvalue of these operators. For reducible operators, we prove that the lower spectral radius always serves as the least upper bound of the set of eigenvalues pertaining to positive eigenvectors, and the upper spectral radius the greatest lower bound of the set. Finally, we demonstrate the full power of these Krein-Rutman theorems on some PDE examples such as elliptic eigenvalue problems on non-smooth domains, and cooperative systems which may or may not be fully coupled, by using as few PDE tools as possible.

On the index of convergence of a class of Boolean matrices with structural properties

ABSTRACT Boolean matrices are of prime importance in the study of discrete event systems (DES), which allow us to model systems across a variety of applications. The index of convergence (i.e. the number of distinct powers of a Boolean matrix) is a crucial characteristic in that it assesses the transient behaviour of the system until reaching a periodic course. In this paper, adopting a graph-theoretic approach, we present bounds for the index of convergence of Boolean matrices for a diverse class of systems, with a certain decomposition. The presented bounds are an extension of the bound on irreducible Boolean matrices, and we provide non-trivial bounds that were unknown for classes of systems. Furthermore, the proposed method is able to determine the bounds in polynomial time. Lastly, we illustrate how the new bounds compare with the previously known bounds and we show their effectiveness in cases such as the benchmark IEEE 5-bus power system.