Nonnegative Eigenvector Research Articles

Finding the global optimum of a class of quartic minimization problem

We consider a special nonconvex quartic minimization problem over a single spherical constraint, which includes the discretized energy functional minimization problem of non-rotating Bose-Einstein condensates (BECs) as one of the important applications. Such a problem is studied by exploiting its characterization as a nonlinear eigenvalue problem with eigenvector nonlinearity (NEPv). Firstly, we show that the NEPv has a unique nonnegative eigenvector, corresponding to the smallest nonlinear eigenvalue of NEPv, which is exactly the global minimizer to the optimization problem. Secondly, with these properties, we obtain that any algorithm converging to the nonnegative stationary point of this optimization problem finds its global optimum, such as the regularized Newton method. In particular, we obtain the convergence to the global optimum of the inexact alternating direction method of multipliers for this problem. Numerical experiments for applications in non-rotating BECs validate our theories.

The geometry connectivity of hypergraphs

Let G be a k-uniform hypergraph, and LG be its Laplacian tensor. The β(G) denotes the maximum number of linearly independent nonnegative eigenvectors of LG corresponding to the eigenvalue 0. In this paper, β(G) is called the geometry connectivity of G. We show that the number of connected components of G equals the geometry connectivity β(G).

Nonnegative Eigenvectors of Symmetric Matrices

For matrices with all nonnegative entries, the Perron–Frobenius theorem guarantees the existence of an eigenvector with all nonnegative components. We show that the existence of such an eigenvector is also guaranteed for a very different class of matrices, namely real symmetric matrices with exactly two eigenvalues. We also prove a partial converse, that among real symmetric matrices with any more than two eigenvalues there exist some having no nonnegative eigenvector.

The eigenvectors to the p-spectral radius of general hypergraphs

Let $$\mathcal {A}$$ be the adjacency tensor of a general hypergraph H. For any real number $$p\ge 1$$ , the p-spectral radius $$\lambda ^{(p)}(H)$$ of H is defined as $$\lambda ^{(p)}(H)=\max \{x^{\mathrm {T}}(\mathcal {A}x)\,|\,x\in {\mathbb {R}}^n, \Vert x\Vert _p=1\}$$ . In this paper we present some bounds on entries of the nonnegative unit eigenvector corresponding to the p-spectral radius of H, which generalize the relevant results of uniform hypergraphs/graphs in the literature.

Aggregating nonnegative eigenvectors of the adjacency matrix as a measure of centrality for a directed graph

ABSTRACTEigenvector centrality is a popular measure that uses the principal eigenvector of the adjacency matrix to distinguish importance of nodes in a graph. To find the principal eigenvector, the power method iterating from a random initial vector is often adopted. In this article, we consider the adjacency matrix of a directed graph and choose suitable initial vectors according to strongly connected components of the graph instead so that nonnegative eigenvectors, including the principal one, can be found. Consequently, for aggregating nonnegative eigenvectors, we propose a weighted measure of centrality, called the aggregated-eigenvector centrality. Weighting each nonnegative eigenvector by the reachability of the associated strongly connected component, we can obtain a measure that follows a status hierarchy in a directed graph.

Cost sharing solutions defined by non-negative eigenvectors

Financial support from Spanish Ministry of Economy and Competitiveness under Project ECO2013-43119 is gratefully acknowledged. Silva-Reus also acknowledges financial support from the Generalitat Valenciana, Spain under Project PROMETEO/2009/068.

Balanced Growth Solutions and Related Problems of Hua's Macroeconomic Model

In this paper, the balanced growth solution and relative stability of solution on Hua's macroeconomic model is studied. Firstly, by deriving the greatest eigenvalue and nonnegative eigenvector, and analyzing the range of the eigenvalue of nonnegative matrix, the existence of the balanced growth solution on a sort of Hua's macroeconomic model is proved, no matter whether direct consumption coefficient matrix is irreducible or reducible. Furthermore, the concept of balanced solution's relative stability is introduced to Hua's macroeconomic model and the necessary and sufficient condition for the existence of these solutions is obtained. Finally, based on the price equation proposed by Prof. Hua, the dynamic price system which is inclusive of the interest rate is proposed and the relationship between price and output on the basis of the relative stability of price system is illuminated. Index Terms—Hua's macroeconomic model, balanced growth solution, relative stability, price system In the 1980s,Prof. Luogeng Hua established the well- known theory which called mathematical theory of large-scale optimization in planned economy in several papers. In these papers, he proposed the so-called Hua's macroeconomic model. Under the condition of enough large productivity elasticity, the model depicts the relationship between input and output. Compared with Leontief's macroeconomic model, Hua's model fits better in the current Chinese economy. Hence the model can be used to analyze and forecast effectively in China

Two cores of a nonnegative matrix

We prove that the sequence of eigencones (i.e., cones of nonnegative eigenvectors) of positive powers Ak of a nonnegative square matrix A is periodic both in max algebra and in nonnegative linear algebra. Using an argument of Pullman, we also show that the Minkowski sum of the eigencones of powers of A is equal to the core of A defined as the intersection of nonnegative column spans of matrix powers, also in max algebra. Based on this, we describe the set of extremal rays of the core.The spectral theory of matrix powers and the theory of matrix core is developed in max algebra and in nonnegative linear algebra simultaneously wherever possible, in order to unify and compare both versions of the same theory.

The Laplacian of a uniform hypergraph

In this paper, we investigate the Laplacian, i.e., the normalized Laplacian tensor of a $$k$$ -uniform hypergraph. We show that the real parts of all the eigenvalues of the Laplacian are in the interval $$[0,2]$$ , and the real part is zero (respectively two) if and only if the eigenvalue is zero (respectively two). All the H $$^+$$ -eigenvalues of the Laplacian and all the smallest H $$^+$$ -eigenvalues of its sub-tensors are characterized through the spectral radii of some nonnegative tensors. All the H $$^+$$ -eigenvalues of the Laplacian that are less than one are completely characterized by the spectral components of the hypergraph and vice verse. The smallest H-eigenvalue, which is also an H $$^+$$ -eigenvalue, of the Laplacian is zero. When $$k$$ is even, necessary and sufficient conditions for the largest H-eigenvalue of the Laplacian being two are given. If $$k$$ is odd, then its largest H-eigenvalue is always strictly less than two. The largest H $$^+$$ -eigenvalue of the Laplacian for a hypergraph having at least one edge is one; and its nonnegative eigenvectors are in one to one correspondence with the flower hearts of the hypergraph. The second smallest H $$^+$$ -eigenvalue of the Laplacian is positive if and only if the hypergraph is connected. The number of connected components of a hypergraph is determined by the H $$^+$$ -geometric multiplicity of the zero H $$^+$$ -eigenvalue of the Lapalacian.

A link between the Perron–Frobenius theorem and Perron’s theorem for difference equations

Let A n , n ∈ N , be a sequence of k × k matrices which converge to a matrix A as n → ∞ . It is shown that if x n , n ∈ N , is a sequence of nonnegative nonzero vectors such that x n + 1 = A n x n , n ∈ N , then ρ = lim n → ∞ ‖ x n ‖ n is an eigenvalue of the limiting matrix A with a nonnegative eigenvector. This result implies the weak form of the Perron–Frobenius theorem and for the class of nonnegative solutions it improves the conclusion of a Perron type theorem for difference equations.

An index theory for uniformly locally finite graphs

An index theory for uniformly locally finite (ULF) graphs is developed based on the adjacency operator A acting on the space of bounded sequences defined on the vertices. It turns out that the characterization by upper and lower nonnegative eigenvectors is an appropriate tool to overcome the difficulties imposed by the ℓ ∞ -setting. A distinctive property of the spectral radius r ∞ ( A ) in ℓ ∞ is the identity r ∞ = sup λ ⩾ 0 ∃ x ∈ ℓ ∞ Γ , x > 0 : A x ⩾ λ x = : I , whilethe ℓ 2 -spectral radius r 2 of the adjacency operator satisfies r 2 = inf λ ⩾ 0 ∃ x ∈ ℓ ∞ Γ , x > 0 : A x ⩽ λ x . The index I , as well as other order indices, can serve in classifying ULF graphs and enables connections with various graph invariants. E.g., the chromatic number can be estimated from above by 1 + r ∞ . Moreover, results on the index I in the periodic case, the regular one and for graphs having only finitely many essential ramification nodes are presented.

On general matrices having the Perron-Frobenius Property

A matrix is said to have the Perron-Frobenius property if its spectral radius is aneigenvalue with a corresponding nonnegative eigenvector. Matrices having this and similar properties are studied in this paper as generalizations of nonnegative matrices. Sets consisting of such generalized nonnegative matrices are studied and certain topological aspects such as connectedness and closure are proved. Similarity transformations leaving such sets invariant are completely described, and it is shown that a nonnilpotent matrix eventually capturing the Perron-Frobenius propertyis in fact a matrix that already has it.

K-THEORETIC INVARIANTS AND CONFORMAL MEASURES OF THE DYCK SHIFTS

The K-groups and the Bowen–Franks groups for subshifts are invariant under not only topological conjugacy but also flow equivalence. In this paper, the K-groups and the Bowen–Franks groups for the Dyck shift D2with 4 brackets are calculated. The spectrum of D2with bounded nonnegative eigenvectors is shown to be the singleton {3}. We then prove that a unique conformal probability measure on the right one-sided subshift XD2exists. These results are extended to general Dyck shifts DNwith 2N brackets.

Spectral theory of copositive matrices

Let A ∈ R n× n . We provide a block characterization of copositive matrices, with the assumption that one of the principal blocks is positive definite. Haynsworth and Hoffman showed that if r is the largest eigenvalue of a copositive matrix then r ⩾ ∣ λ∣, for all other eigenvalues λ of A. We continue their study of the spectral theory of copositive matrices and show that a copositive matrix must have a positive vector in the subspace spanned by the eigenvectors corresponding to the nonnegative eigenvalues. Moreover, if a symmetric matrix has a positive vector in the subspace spanned by the eigenvectors corresponding to its nonnegative eigenvalues, then it is possible to increase the nonnegative eigenvalues to form a copositive matrix A′, without changing the eigenvectors. We also show that if a copositive matrix has just one positive eigenvalue, and n − 1 nonpositive eigenvalues then A has a nonnegative eigenvector corresponding to a nonnegative eigenvalue.

Existence of quasi-stationary measures for asymmetric attractive particle systems on $\ZZ^d$

We show the existence of nontrivial quasi-stationary measures for conservative attractive particle systems on $\ZZ^d$ conditioned on avoiding an increasing local set $\A$. Moreover, we exhibit a sequence of measures $\{\nu_n\}$, whose $\omega$-limit set consists of quasi-stationary measures. For zero-range processes, with stationary measure $\nur$, we prove the existence of an $L^2(\nur)$ nonnegative eigenvector for the generator with Dirichlet boundary on $\A$, after establishing a priori bounds on the $\{\nu_n\}$.

Eigenvectors and ratio limit theorems for Markov chains and their relatives

Associated to classes of countable discrete Markov chains or, more generally, column-finite nonnegative infinite matrices, and a finite subset of the state space, is a dimension group. In many cases, this dimension group gives information about the nonnegative eigenvectors of the process. Moreover, the study of the nonnegative eigenvectors is, equivalent to the traces on an analytic one parameter family of dimension groups. We pay particular attention to the case that there is at most one nonnegative eigenvector per eigenvalue, giving a number of sufficient conditions. Using the techniques developed here, we also show that under a reasonable set of conditions (principle among them that there be just one nonnegative eigenvector for the spectral radius), a (one-sided) ratio limit theorem holds.

On nonnegative operators and fully cyclic peripheral spectrum

In this note the properties of the peripheral spectrumof a nonnegative linear operator A (for which the spectral radius is a pole of its resolvent) in a complex Banach lattice are studied. It is shown, e.g., that the peripheral spectrum of a natural quotient operator is always fully cyclic. We describe when the nonnegative eigenvectors corresponding to the spectral radius r span the kernel N(r A). Finally, we apply our results to the case of a nonnegative matrix, and show that they sharpen earlier results by B.-S. Tam [Tamkang J. Math. 21:65{70, 1990] on such matrices and full cyclicity of the peripheral spectrum. AMS subject classi cations. 47B65, 47A10, 15A48

Z-Pencils

The matrix pencil (A,B) = {tB-A | t \in C} is considered under the assumptions that A is entrywise nonnegative and B-A is a nonsingular M-matrix. As t varies in [0,1], the Z-matrices tB-A are partitioned into the sets L_s introduced by Fiedler and Markham. As no combinatorial structure of B is assumed here, this partition generalizes some of their work where B=I. Based on the union of the directed graphs of A and B, the combinatorial structure of nonnegative eigenvectors associated with the largest eigenvalue of (A,B) in [0,1) is considered.

Asymptotic properties of powers of nonnegative matrices, with applications

Let A be a finite square (reducible) nonnegative matrix. The main theorem of the paper gives first order approximations of f n ( A), for certain sequences f n of analytic functions. In particular, the theorem holds when f n ( A)= A n . The theorem is applied to study the local behavior of large powers of a nonnegative matrix, to study the limiting output vectors of a nonnegative multiplicative process, and to characterize the nonnegative eigenvectors of a nonnegative matrix. Finally, an application to absorbing Markov chains is given.