<Emphasis Type="Italic">l</Emphasis><Superscript><Emphasis Type="Italic">k</Emphasis>,<Emphasis Type="Italic">s</Emphasis></Superscript>-Singular values and spectral radius of partially symmetric rectangular tensors

The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we first study properties of l ^k,s -singular values of real rectangular tensors. Then, a necessary and sufficient condition for the positive definiteness of partially symmetric rectangular tensors is given. Furthermore, we show that the weak Perron-Frobenius theorem for nonnegative partially symmetric rectangular tensor keeps valid under some new conditions and we prove a maximum property for the largest l ^k,s -singular values of nonnegative partially symmetric rectangular tensor. Finally, we prove that the largest l ^k,s -singular value of nonnegative weakly irreducible partially symmetric rectangular tensor is still geometrically simple.     

Singular values of nonnegative rectangular tensors

The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. Some properties concerning the singular values of a real rectangular tensor were discussed by K. C. Chang et al. [J. Math. Anal. Appl., 2010, 370: 284–294]. In this paper, we give some new results on the Perron-Frobenius Theorem for nonnegative rectangular tensors. We show that the weak Perron-Frobenius keeps valid and the largest singular value is really geometrically simple under some conditions. In addition, we establish the convergence of an algorithm proposed by K. C. Chang et al. for finding the largest singular value of nonnegative primitive rectangular tensors.     

Standard tensor and its applications in problem of singular values of tensors

In this paper, first we give the definition of standard tensor. Then we clarify the relationship between weakly irreducible tensors and weakly irreducible polynomial maps by the definition of standard tensor. And we prove that the singular values of rectangular tensors are the special cases of the eigen-values of standard tensors related to rectangular tensors. Based on standard tensor, we present a generalized version of the weak Perron-Frobenius Theorem of nonnegative rectangular tensors under weaker conditions. Furthermore, by studying standard tensors, we get some new results of rectangular tensors. Besides, by using the special structure of standard tensors corresponding to nonnegative rectangular tensors, we show that the largest singular value is really geometrically simple under some weaker conditions.     

Real eigenvalues of nonnegative matrices which commute with a symmetric matrix involution

It is well known that if P is a nonnegative matrix, then its spectral radius is an eigenvalue of P (Perron-Frobenius theorem). In this paper it is shown that if P is an n × n nonnegative matrix and it commutes with a nonnegative symmetric involution when n=4m+3, then (1) P has at least two real eigenvalues if n=4m or 4m + 2, (2) P has at least one real eigenvalue if n=4m+1, and (3) P has at least three real eigenvalues if n=4m+3, where m is a nonnegative integer and n ? 1. Examples are given to show that these results are the best possible, and nonnegative symmetric involutions are classified.     

关于非负张量Z-谱半径的一些结果（英文）

Z-eigenvalue plays a fundamental role in the best rank-one approximation.Chang,Pearson,and Zhang generalized the Perron-Probenius theorem for Z-eigenvalues of nonnegative tensors and gave some properties of the Z-spectral radius recently.In this paper,we give some properties of Z-eigenvectors associated with Z-spectral radius of nonnegative weakly symmetric tensors,compare the Zspectral radius between two nonnegative tensors,and modify the upper and lower bounds for Z-spectral radius.Some results for the Z-singular values of rectangular tensors are also given.     

Strictly nonnegative tensors and nonnegative tensor partition

We introduce a new class of nonnegative tensors—strictly nonnegative tensors.A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa.We show that the spectral radius of a strictly nonnegative tensor is always positive.We give some necessary and su?cient conditions for the six wellconditional classes of nonnegative tensors,introduced in the literature,and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors.We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility.We show that for a nonnegative tensor T,there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible;and the spectral radius of T can be obtained from those spectral radii of the induced tensors.In this way,we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption.Some preliminary numerical results show the feasibility and effectiveness of the algorithm.     

Singular values of a real rectangular tensor

Real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we systematically study properties of singular values of a real rectangular tensor, and give an algorithm to find the largest singular value of a nonnegative rectangular tensor. Numerical results show that the algorithm is efficient.     

Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming

Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, for a tensor with nonnegative entries (which we refer it as a nonnegative tensor), efficient numerical schemes have been proposed to calculate its maximum eigenvalue based on a Perron–Frobenius-type theorem. In this paper, we consider a new class of tensors called essentially nonnegative tensors, which extends the concept of nonnegative tensors, and examine the maximum eigenvalue of an essentially nonnegative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially nonnegative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in its turn, can be equivalently rewritten as a semi-definite programming problem. Then, using this sum of squares programming problem, we also provide upper and lower estimates for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor. Numerical examples are also presented to illustrate the significance of the results.     

Geometric conditions for the existence of positive eigenvalues of matrices

Finite-dimensional theorems of Perron-Frobenius type are proved. For A∈$\text{C}$ⁿⁿ and a nonnegative integer k, we let w_k (A) be the cone generated by A^k, A^k+1,…in $\text{C}$ⁿⁿ. We show that A satisfies the Perron-Schaefer condition if and only if the closure $\text{W}$_k(A) of w_k(A) is a pointed cone. This theorem is closely related to several known results. If k?v₀(A), the index of the eigenvalue 0 in spec A, we prove that A has a positive eigenvalue if and only if w_k(A) is a pointed nonzero cone or, equivalently $\text{W}$_k(A) is not a real subspace of $\text{C}$ⁿⁿ. Our proofs are elementary and based on a method of Birkhoff's. We discuss the relation of this method to Pringsheim's theorem.     

On computing minimal <Emphasis Type="Italic">H</Emphasis>-eigenvalue of sign-structured tensors

Finding the minimal H-eigenvalue of tensors is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a sum-of-squares (SOS) algorithm for computing the minimal H-eigenvalues of tensors with some sign structures called extended essentially nonnegative tensors (EEN-tensors), which includes nonnegative tensors as a subclass. In the even-order symmetric case, we first discuss the positive semi-definiteness of EEN-tensors, and show that a positive semi-definite EEN-tensor is a nonnegative tensor or an M-tensor or the sum of a nonnegative tensor and an M-tensor, then we establish a checkable sufficient condition for the SOS decomposition of EEN-tensors. Finally, we present an efficient algorithm to compute the minimal H-eigenvalues of even-order symmetric EEN-tensors based on the SOS decomposition. Numerical experiments are given to show the efficiency of the proposed algorithm.     

Structured nonnegative matrix factorization with applications to hidden Markov realization and clustering

In this paper, we study the structured nonnegative matrix factorization problem: given a square, nonnegative matrix P, decompose it as P=VAV^? with V and A nonnegative matrices and with the dimension of A as small as possible. We propose an iterative approach that minimizes the Kullback-Leibler divergence between P and VAV^? subject to the nonnegativity constraints on A and V with the dimension of A given. The approximate structured decomposition P?VAV^? is closely related to the approximate symmetric decomposition P?VV^?. It is shown that the approach for finding an approximate structured decomposition can be adapted to solve the symmetric decomposition problem approximately. Finally, we apply the nonnegative decomposition VAV^? to the hidden Markov realization problem and to the clustering of data vectors based on their distance matrix.     

Metrics and undirected cuts

Suppose thatG is an undirected graph whose edges have nonnegative integer-valued lengthsl(e), and that {s ₁,t ₁},?, {s _m ,t _m } are pairs of its vertices. Can one assign nonnegative weights to the cuts ofG such that, for each edgee, the total weight of cuts containinge does not exceedl(e) and, for eachi, the total weight of cuts ‘separating’s _i andt _i is equal to the distance (with respect tol) betweens _i andt _i ? Using linear programming duality, it follows from Papernov's multicommodity flow theorem that the answer is affirmative if the graph induced by the pairs {s ₁,t ₁},?, {s _m ,t _m } is one of the following: (i) the complete graph with four vertices, (ii) the circuit with five vertices, (iii) a union of two stars. We prove that if, in addition, each circuit inG has an even length (with respect tol) then there exists a suitable weighting of the cuts with the weights integer-valued; moreover, an algorithm of complexity O(n ³) (n is the number of vertices ofG) is developed for solving such a problem. Also a class of metrics decomposable into a nonnegative linear combination of cut-metrics is described, and it is shown that the separation problem for cut cones isNP-hard.     

Maximal number of distinct H-eigenpairs for a two-dimensional real tensor

Based on the generalized characteristic polynomial introduced by J. Canny in Generalized characteristic polynomials [J. Symbolic Comput., 1990, 9(3): 241–250], it is immediate that for any m-order n-dimensional real tensor, the number of distinct H-eigenvalues is less than or equal to n(m?1) ^n?1. However, there is no known bounds on the maximal number of distinct Heigenvectors in general. We prove that for any m ? 2, an m-order 2-dimensional tensor A exists such that A has 2(m ? 1) distinct H-eigenpairs. We give examples of 4-order 2-dimensional tensors with six distinct H-eigenvalues as well as six distinct H-eigenvectors. We demonstrate the structure of eigenpairs for a higher order tensor is far more complicated than that of a matrix. Furthermore, we introduce a new class of weakly symmetric tensors, called p-symmetric tensors, and show under certain conditions, p-symmetry will effectively reduce the maximal number of distinct H-eigenvectors for a given two-dimensional tensor. Lastly, we provide a complete classification of the H-eigenvectors of a given 4-order 2-dimensional nonnegative p-symmetric tensor. Additionally, we give sufficient conditions which prevent a given 4-order 2-dimensional nonnegative irreducible weakly symmetric tensor from possessing six pairwise distinct H-eigenvectors.     

Expansion of a scalar function on the unit sphere sn−1 in terms of tensor components

It is shown that eachl-th partial sum of the Fourier series for a scalar function on the unit sphere S^n?1 can be expressed as a polynomial in the components of two completely symmetric tensors of rankl andl?1, respectively, provided thatl≥1. It is proposed that this expansion should be used to describe the so-called limiting surfaces in second rank tensor spaces.     

On the optimal computation of a set of symmetric and persymmetric bilinear forms

A special class $\text{T}$ⁿ of n×n matrices is described, which has tensor rank n over the real field. A tensor base for general symmetric, persymmetric, both symmetric and persymmetric matrices and Toeplitz symmetric matrices can be defined in terms of the tensor bases of $\text{T}$^l for some different values of l. It is proved that both symmetric and persymmetric n×n matrices and Toeplitz symmetric n×n matrices have tensor rank $\text{[}\text{(n+1)}{}^2\text{4}\text{]}$ and 2n?2, respectively, in the real field.     

Eigenvalue problem for finite difference equations with p-Laplacian

In this paper, we consider a discrete four-point boundary value problem $$\triangle\bigl(\phi_p\bigl(\triangle u(k-1)\bigr)\bigr)+ \lambda e(k)f\bigl(u(k)\bigr)=0,\quad k\in N(1,T),$$ subject to boundary conditions $$\triangle u(0)-\alpha u(l_{1})=0,\qquad\triangle u(T)+\beta u(l_{2})=0,$$ by a simple application of a fixed point theorem. If e(k),f(u(k)) are nonnegative, the solutions of the above problem may not be nonnegative, this is the main difficulty for us to study positive solution of this problem. In this paper, we give restrictive conditions ??l ₁??1, ??(T+1?l ₂)??1 to guarantee the solutions of this problem are nonnegative, if it has, under the conditions e(k),f(u(k)) are nonnegative. We first construct a new operator equation which is equivalent to the problem and provide sufficient conditions for the nonexistence and existence of at least one or two positive solutions. In doing so, the usual restrictions $f_{0}=\lim_{u\rightarrow 0^{+}}\frac{f(u)}{\phi_{p}(u)}$ and $f_{\infty}=\lim_{u\rightarrow\infty}\frac{f(u)}{\phi_{p}(u)}$ exist are removed.     

Global behavior of a higher order difference equation

The aim of this paper is to investigate the global stability and periodic nature of the positive solutions of the difference equation $$x_{n + 1} = \frac{{A + Bx_{n - 2k - 1} }} {{C + D\prod\limits_{i = 1}^k {x_{n - 2i} } }}, n = 0,1,2, \ldots ,$$ where A, B are nonnegative real numbers, C,D > 0 and l, k are nonnegative integers such that l ≤ k.     

The nonlinear complementarity problem with applications,part 1

The main result in this paper is an existence and uniqueness theorem for the following nonlinear complementarity problem: Given a mapping from then-dimensional Euclidean spaceE ⁿ into itself, find a nonnegative vector inE ⁿ whose image, under the given mapping, is also nonnegative, the two vectors being orthogonal to each other. It is shown that the above problem has a unique solution if the given mapping is continuous and strongly monotone on the nonnegative orthantE ₊ ⁿ ofE ⁿ . It is also shown that a sufficient condition for a differentiable mapping to be strongly monotone on an open set is that all the eigenvalues of the symmetric part of its Jacobian be bounded below by a positive constant on the given set.This research constituted a part of the author's Ph.D. dissertation at the University of California at Berkeley, Berkeley, California. The author would like to express his appreciation to Professor G. B. Dantzig, who brought this problem to his attention and guided his research with his several suggestions and helpful criticism. Also, he thanks the referee for several important comments and recommendations.     

Spectral properties of odd-bipartite <Emphasis Type="Italic">Z</Emphasis>-tensors and their absolute tensors

Stimulated by odd-bipartite and even-bipartite hypergraphs, we define odd-bipartite (weakly odd-bipartie) and even-bipartite (weakly evenbipartite) tensors. It is verified that all even order odd-bipartite tensors are irreducible tensors, while all even-bipartite tensors are reducible no matter the parity of the order. Based on properties of odd-bipartite tensors, we study the relationship between the largest H-eigenvalue of a Z-tensor with nonnegative diagonal elements, and the largest H-eigenvalue of absolute tensor of that Ztensor. When the order is even and the Z-tensor is weakly irreducible, we prove that the largest H-eigenvalue of the Z-tensor and the largest H-eigenvalue of the absolute tensor of that Z-tensor are equal, if and only if the Z-tensor is weakly odd-bipartite. Examples show the authenticity of the conclusions. Then, we prove that a symmetric Z-tensor with nonnegative diagonal entries and the absolute tensor of the Z-tensor are diagonal similar, if and only if the Z-tensor has even order and it is weakly odd-bipartite. After that, it is proved that, when an even order symmetric Z-tensor with nonnegative diagonal entries is weakly irreducible, the equality of the spectrum of the Z-tensor and the spectrum of absolute tensor of that Z-tensor, can be characterized by the equality of their spectral radii.     

On discrete ℓ1-regularization

A real m × n matrix A and a vector y?∈?? ^m determine the discrete l ¹-regularization (DLR) problem 0.1 $$ \min \left\{\mbox{\,}|y-Ax|_1+\rho |x|_1:\,x\in\mathbb{R}^n \right\}, $$ where | · |₁ denotes the l ¹-norm of a vector and ρ?≥?0 is a nonnegative parameter. In this paper, we provide a detailed analysis of this problem which include a characterization of all solutions to (0.1), remarks about the geometry of the solution set and an effective iterative algorithm for numerical solution of (0.1). We are specially interested in the behavior of the solution of (0.1) as a function of ρ and in this regard, we prove in general the existence of critical values of ρ between which the l ¹-norm of any solution remains constant. These general remarks are significantly refined when A is a strictly totally positive (STP) matrix. The importance of STP matrices is well-established [5, 14]. Under this setting, the relationship between the number of nonzero coordinates of a distinguished solution of the DLR problem is precisely explained as a function of the regularization parameter for a certain class of vectors in ? ^m . Throughout our analysis of the DLR problem, we emphasize the importance of the dual maximum problem by demonstrating that any solution of it leads to a solution of the DLR problem, and vice versa.