Nonnegative Irreducible Matrices Research Articles

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.

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.

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.

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.

Perturbation analysis in thermodynamics using matrix representations of Ruelle operators and its application to graph IFS

We study perturbations of topological pressures, Gibbs measures and measure-theoretic entropies of these measures concerning perturbed potentials defined on topologically transitive subshift of finite type. The subshift with respect to non-perturbed system is not assumed to be topologically transitive. Therefore, the subshift of the perturbed systems and the subshift of the unperturbed system are different. We reduce this situation to a perturbation problem of certain irreducible nonnegative matrices generated by Ruelle operators. Moreover, under suitable conditions of potentials, we characterize the limit points of those thermodynamics and give a necessary and sufficient condition for convergence of Gibbs measures and the measure-theoretic entropy of this measure when the subshift of the non-perturbed system has or transitive components with the maximal pressure. Finally we apply our results to a problem of collapse of perturbed graph iterated function systems.

A Necessary and Sufficient Condition for Existence of a Positive Perron Vector

In 1907, Perron showed that a positive square matrix has a unique largest positive eigenvalue with a positive eigenvector. This result was extended to irreducible nonnegative matrices by Frobenius in 1912, and to irreducible nonnegative tensors and weakly irreducible nonnegative tensors recently. This result is a fundamental result in matrix theory and has found wide applications in probability theory, internet search engines, spectral graph and hypergraph theory, etc. In this paper, we give a necessary and sufficient condition for the existence of such a positive eigenvector, i.e., a positive Perron vector, for a nonnegative tensor. We show that every nonnegative tensor has a canonical nonnegative partition form, from which we introduce strongly nonnegative tensors. A tensor is called strongly nonnegative if the spectral radius of each genuine weakly irreducible block is equal to the spectral radius of the tensor, which is strictly larger than the spectral radius of any other block. We prove that a nonne...

Geometric mapping properties of semipositive matrices

Semipositive matrices map a positive vector to a positive vector and as such they are a very broad generalization of the irreducible nonnegative matrices. Nevertheless, the ensuing geometric mapping properties of semipositive matrices result in several parallels to the theory of cone preserving and cone mapping matrices. It is shown that for a semipositive matrix A, there exist a proper polyhedral cone K1 of nonnegative vectors and a polyhedral cone K2 of nonnegative vectors such that AK1=K2. The set of all nonnegative vectors mapped by A to the nonnegative orthant is a proper polyhedral cone; as a consequence, A belongs to a proper polyhedral cone comprising semipositive matrices. When the powers Ak (k=0,1,…) have a common semipositivity vector, then A has a positive eigenvalue. If A has a sole peripheral eigenvalue λ and the powers of A have a common semipositivity vector with a non-vanishing term in the direction of the left eigenspace of λ, then A leaves a proper cone invariant.

Matrix roots of imprimitive irreducible nonnegative matrices

Using matrix function theory, Perron–Frobenius theory, combinatorial matrix theory, and elementary number theory, we characterize, classify, and describe in terms of the Jordan canonical form the matrix pth-roots of imprimitive irreducible nonnegative matrices. Preliminary results concerning the matrix roots of reducible matrices are provided as well.

Bounds for the Perron Root of Irreducible Nonnegative Matrices

Some bounds for the Perron root $\rho$ of positive matrices are proposed. We proved that$$\max_{1\leq i \leq n}{\bigg(\sum_{k}{a_{ik}m_{ki}} \bigg)} \leq \rho \leq \min_{1\leq i \leq n}{\bigg(\sum_{k}{a_{ik}M_{ki}} \bigg)}.$$where $$m_{ki}=\min_{t}{\frac {(\alpha_k,\beta_t)}{(\alpha_i,\beta_t)}}, M_{ki}=\max_{t}{\frac {(\alpha_k,\beta_t)}{(\alpha_i,\beta_t)}}$$and $\alpha_k$ denotes the $k$-th row of matrix $A$, $\beta_t$ the $t$-th column of $A$, $(\alpha_k,\beta_t)$ denotes the inner product of $\alpha_k$ and $\beta_t$.And these bounds can also be used to estimate the Perron root of nonnegative irreducible matrices.

Exact and Heuristic Algorithms for Semi-Nonnegative Matrix Factorization

Given a matrix $M$ (not necessarily nonnegative) and a factorization rank $r$, semi-nonnegative matrix factorization (semi-NMF) looks for a matrix $U$ with $r$ columns and a nonnegative matrix $V$ with $r$ rows such that $UV$ is the best possible approximation of $M$ according to some metric. In this paper, we study the properties of semi-NMF from which we develop exact and heuristic algorithms. Our contribution is threefold. First, we prove that the error of a semi-NMF of rank $r$ has to be smaller than the best unconstrained approximation of rank $r-1$. This leads us to a new initialization procedure based on the singular value decomposition (SVD) with a guarantee on the quality of the approximation. Second, we propose an exact algorithm (that is, an algorithm that finds an optimal solution), also based on the SVD, for a certain class of matrices (including nonnegative irreducible matrices) from which we derive an initialization for matrices not belonging to that class. Numerical experiments illustrate that this second approach performs extremely well, and allows us to compute optimal semi-NMF decompositions in many situations. Finally, we analyze the computational complexity of semi-NMF proving its NP-hardness, already in the rank-one case (that is, for $r = 1$), and we show that semi-NMF is sometimes ill-posed (that is, an optimal solution does not exist).

Testing the irreducibility of nonsquare Perron–Frobenius systems

The Perron–Frobenius (PF) theorem provides a simple characterization of the eigenvectors and eigenvalues of irreducible nonnegative square matrices. A generalization of the PF theorem to nonsquare matrices, which can be interpreted as representing systems with additional degrees of freedom, was recently presented in [1]. This generalized theorem requires a notion of irreducibility for nonsquare systems. A suitable definition, based on the property that every maximal square (legal) subsystem is irreducible, is provided in [1], and is shown to be necessary and sufficient for the generalized theorem to hold. This note shows that irreducibility of a nonsquare system can be tested in polynomial time. The analysis uses a graphic representation of the nonsquare system, termed the constraint graph, representing the flow of influence between the constraints of the system.

Optimization of an amplification protocol for misfolded proteins by using relaxed control

We investigate an optimal control problem which arises in the optimization of an amplification technique for misfolded proteins. The improvement of this technique may play a role in the detection of prion diseases. The model consists in a linear system of differential equations with a nonlinear control. The appearance of oscillations in the numerical simulations is understood by using the Perron and Floquet eigenvalue theory for nonnegative irreducible matrices. Then to overcome the unsolvability of the optimal control, we relax the problem. In the two dimensional case, we solve explicitly the optimal relaxed control problem when the final time is large enough.

KMS states on [formula omitted]-algebras associated to higher-rank graphs

Consider a higher-rank graph of rank k. Both the Cuntz–Krieger algebra and the Toeplitz–Cuntz–Krieger algebra of the graph carry natural gauge actions of the torus Tk, and restricting these gauge actions to one-parameter subgroups of Tk gives dynamical systems involving actions of the real line. We study the KMS states of these dynamical systems. We find that for large inverse temperatures β, the simplex of KMSβ states on the Toeplitz–Cuntz–Krieger algebra has dimension d one less than the number of vertices in the graph. We also show that there is a preferred dynamics for which there is a critical inverse temperature βc: for β larger than βc, there is a d-dimensional simplex of KMS states; when β=βc and the one-parameter subgroup is dense, there is a unique KMS state, and this state factors through the Cuntz–Krieger algebra. As in previous studies for k=1, our main tool is the Perron–Frobenius theory for irreducible non-negative matrices, though here we need a version of the theory for commuting families of matrices.

Probabilistic approach to Perron root, the group inverse, and applications

A probabilistic approach to the study of the Perron root of irreducible nonnegative matrices is presented. Our two main results are a probabilistic representation for the generalized inverse of the generator of a continuous-time finite-state Markov chain and the identification of the Hessian of the Perron root of a nonnegative irreducible matrix, with respect to a certain natural transformation of its entries, as a covariance matrix for additive functionals for a related Markov chain. These provide us with a natural approach – at least from the point of view of the probabilist – to study positivity and convexity properties of perturbations to the Perron eigenvalue. We focus on reestablishing and improving a number of known results in the field.

A new algorithm for computing largest real part eigenvalue of matrices: Collatz & Perron-Frobernius' approach

This paper describes a new method and algorithm for the numerical solution of eigenvalues with the largest real part of positive matrices. The method is based on a numerical implementation of Collatz's eigenvalue inclusion theorem for non-negative irreducible matrices. Eigenvalues are analyzed for the studies of the stability of linear systems. Finally, a numerical discussion is given to derive the required number of mathematical operations of the new algorithm. Comparisons between the new algorithm and several well known ones, such as Power, and QR methods, are discussed.

Max-algebraic attraction cones of nonnegative irreducible matrices

It is known that the max-algebraic powers A r of a nonnegative irreducible matrix are ultimately periodic. This leads to the concept of attraction cone Attr( A, t), by which we mean the solution set of a two-sided system λ t(A)A r⊗ x=A r+t⊗ x, where r is any integer after the periodicity transient T( A) and λ( A) is the maximum cycle geometric mean of A. A question which this paper answers, is how to describe Attr( A, t) by a concise system of equations without knowing T( A). This study requires knowledge of certain structures and symmetries of periodic max-algebraic powers, which are also described. We also consider extremals of attraction cones in a special case, and address the complexity of computing the coefficients of the system which describes attraction cone.

Tridiagonal matrices with nonnegative entries

In this paper, we characterize the nonnegative irreducible tridiagonal matrices and their permutations, using certain entries in their primitive idempotents. Our main result is summarized as follows. Let d denote a nonnegative integer. Let A denote a matrix in R and let { θ dq i } i = 0 d denote the roots of the characteristic polynomial of A. We say A is multiplicity-free whenever these roots are mutually distinct and contained in R . In this case E i will denote the primitive idempotent of A associated with theta i ( 0 ⩽ i ⩽ d ) . We say A is symmetrizable whenever there exists an invertible diagonal matrix Δ ∈ R such that Δ A Δ - 1 is symmetric. Let Γ ( A ) denote the directed graph with vertex set { 0 , 1 , … , d } , where i → j whenever i ≠ j and A ij ≠ 0 . Theorem. Assume that each entry of A is nonnegative. Then the following are equivalent for 0 ≤ s , t ≤ d . (i) The graph Γ ( A ) is a bidirected path with endpoints s , t : s ↔ * ↔ * ↔ ⋯ ↔ * ↔ t . (ii) The matrix A is symmetrizable and multiplicity-free. Moreover the ( s , t ) -entry of E i times ( θ i - θ 0 ) ⋯ ( θ i - θ i - 1 ) ( θ i - θ i + 1 ) ⋯ ( θ i - θ d ) is independent of i for 0 ≤ i ≤ d , and this common value is nonzero. Recently Kurihara and Nozaki obtained a theorem that characterizes the Q -polynomial property for symmetric association schemes. We view the above result as a linear algebraic generalization of their theorem.

A modified algorithm for the Perron root of a nonnegative matrix

An algorithm of diagonal transformation for the Perron root of nonnegative matrices is proposed by Duan and Zhang [F. Duan, K. Zhang, An algorithm of diagonal transformation for Perron root of nonnegative irreducible matrices, Appl. Math. Comput. 175 (2006) 762–772]. This method can be used for all nonnegative irreducible matrices. In this paper, an improved algorithm which is based on this method is proposed. The new algorithm inherits all the above-mentioned advantages of the original algorithm and has higher efficiency. It is testified by numerical testing that the efficiency of the new algorithm is improved greatly.

Survey of power, QR, and oepomo’s iterative methods for solution of largest eigenvalue of essentially positive matrices

Many of the popular methods for the solution of largest eigenvalue of essentially positive irreducible matrices are surveyed with the hope of finding an efficient method suitable for electromagnetic engineering, radiation problems, system identification problems, and solid mechanics. Eigenvalue computations are both fundamental and ubiquitous in computational science and its fast application areas. Some comparisons between several known algorithms, i.e. Power and QR methods, and earlier theory of Oepomo iterative techniques for solving largest eigenvalue of nonnegative irreducible matrices are presented since there is a continuing demand for new algorithm and library software that efficiently utilize and adapt to new applications.