Nonpositive Off-diagonal Elements Research Articles

On almost semimonotone matrices with non-positive off-diagonal elements and their generalizations

In this paper, we examine almost (strictly) semimonotone matrices which are also either Z-matrices or matrices having Property (++) and show that both parts of Wendler's conjecture [22] (2019) hold for these matrices. Another important result of this paper is the characterization of copositive matrices beyond the matrices with non-positive off-diagonal elements. We discuss some interesting properties related to the structure of almost (strictly) semimonotone matrices and also present results pertaining to the existence and multiplicity of solutions to the linear complementarity problem associated with an almost strictly semimonotone matrix.

Exact SDP relaxations for quadratic programs with bipartite graph structures

For nonconvex quadratically constrained quadratic programs (QCQPs), we first show that, under certain feasibility conditions, the standard semidefinite programming (SDP) relaxation is exact for QCQPs with bipartite graph structures. The exact optimal solutions are obtained by examining the dual SDP relaxation and the rank of the optimal solution of this dual SDP relaxation under strong duality. Our results generalize the previous results on QCQPs with sign-definite bipartite graph structures, QCQPs with forest structures, and QCQPs with nonpositive off-diagonal data elements. Second, we propose a conversion method from QCQPs with no particular structure to the ones with bipartite graph structures. As a result, we demonstrate that a wider class of QCQPs can be exactly solved by the SDP relaxation. Numerical instances are presented for illustration.

Reflected Brownian motion in the quarter plane: An equivalence based on time reversal

We consider a semimartingale reflected Brownian motion (SRBM) Z whose state space is the non-negative quarter plane; the apparently more general case of SRBM in a convex wedge can be transformed to the quarter plane by a simple change of variable. The data of the stochastic process Z are a drift vector μ, a covariance matrix Σ, and a 2 × 2 reflection matrix R whose columns are the directions of reflection on the two axes. We consider only the case where R has non-positive off-diagonal elements, that is, the direction of reflection is either normal or toward the origin from each axis.Under that restriction, we define a dual RBM Zˆ that is constructed using the same data (μ,Σ,R) but with certain sign reversals. Using a time reversal argument, we show the following: to find the distribution of the random two-vector Z(t) at an arbitrary time t, assuming that Z(0)=0, it suffices to find the probability that the dual RBM Zˆ, starting from an arbitrary point z in the quarter plane, reaches the origin before time t. Letting t→∞ and assuming that μ and R jointly satisfy the known condition for positive recurrence of Z, we then have the following: to determine the stationary distribution of Z, it suffices to determine the probability that the dual RBM Zˆ, starting from an arbitrary point z, ever reaches the origin.

Establishing simple relationship between eigenvector and matrix elements

A simple approximate relationship between the ground-state eigenvector and the sum of matrix elements in each row has been established for real symmetric matrices with non-positive off-diagonal elements. Specifically, the i-th components of the ground-state eigenvector could be calculated by a(−Si)p+c, where Si is the sum of elements in the i-th row of the matrix with p, a and c being variational parameters. The simple relationship provides a straightforward method to directly calculate the ground-state eigenvector for a matrix. Our preliminary applications to the Hubbard model and the Ising model in a transverse field show encouraging results. The simple relationship also provides the optimal initial state for other more accurate methods, such as the Lanczos method.

A new strategy for directly calculating the minimum eigenvector of matrices without diagonalization

The diagonalization of matrices may be the top priority in the application of modern physics. In this paper, we numerically demonstrate that, for real symmetric random matrices with non-positive off-diagonal elements, a universal scaling relationship between the eigenvector and matrix elements exists. Namely, each element of the eigenvector of ground states linearly correlates with the sum of matrix elements in the corresponding row. Although the conclusion is obtained based on random matrices, the linear relationship still keeps for non-random matrices, in which off-diagonal elements are non-positive. The relationship implies a straightforward method to directly calculate the eigenvector of ground states for one kind of matrices. The tests on both Hubbard and Ising models show that, this new method works excellently.

Distance–regular graphs having the M-property

We analyse when the Moore–Penrose inverse of the combinatorial Laplacian of a distance–regular graph is an M-matrix; that is, it has non-positive off-diagonal elements or, equivalently when the Moore–Penrose inverse of the combinatorial Laplacian of a distance–regular graph is also the combinatorial Laplacian of another network. When this occurs we say that the distance–regular graph has the M-property. We prove that only distance–regular graphs with diameter up to three can have the M-property and we give a characterization of the graphs that satisfy the M-property in terms of their intersection array. Moreover, we exhaustively analyse strongly regular graphs having the M-property and we give some families of distance–regular graphs with diameter three that satisfy the M-property. Roughly speaking, we prove that all distance–regular graphs with diameter one; about half of the strongly regular graphs; only some imprimitive distance–regular graphs with diameter three, and no distance–regular graphs with diameter greater than three, have the M-property. In addition, we conjecture that no primitive distance–regular graph with diameter three has the M-property.

Triangular and skew-symmetric splitting method for numerical solutions of Markov chains

In this paper, a theorem is presented to indicate that there exists a nonnegative constant ϵ ≥ 0 such that the matrix A = Q T + ϵ I is a positive-definite matrix, where I ∈ R n × n is an identity matrix and Q T ∈ R n × n is a matrix with positive diagonal and nonpositive off-diagonal elements. Then a class of triangular and skew-symmetric splitting (TSS) iteration method is applied to solve the positive-definite linear system A x = b for obtaining the stationary probability vector of an irreducible Markov chain. Theoretical analyses show that the TSS iteration method converges unconditionally to the unique solution of the linear system, with the upper bound of its contraction factor dependent only on the spectrum of the triangular part and independent of the eigenvectors of the matrices involved. Moreover, the inexact triangular and skew-symmetric splitting (ITSS) iteration method, which employs certain Krylov subspace methods as the inner iteration processes at each step of the outer TSS iteration method, is proposed to accelerate the convergence of the TSS iteration method. Numerical experiments are used to illustrate the effectiveness of the TSS and ITSS iteration methods.

Complexity of Stoquastic Frustration-Free Hamiltonians

We study several problems related to properties of nonnegative matrices that arise at the boundary between quantum and classical probabilistic computation. Our results are twofold. First, we identify a large class of quantum Hamiltonians describing systems of qubits for which the adiabatic evolution can be efficiently simulated on a classical probabilistic computer. These are stoquastic local Hamiltonians with a “frustration-free” ground-state. A Hamiltonian belongs to this class iff it can be represented as $H=\sum_{a}H_{a}$ where (1) every term $H_{a}$ acts nontrivially on a constant number of qubits, (2) every term $H_{a}$ has real nonpositive off-diagonal matrix elements in the standard basis, and (3) the ground-state of $H$ is a ground-state of every term $H_{a}$. Second, we generalize the Cook-Levin theorem proving NP-completeness of the satisfiability problem to the complexity class MA (Merlin-Arthur games)—a probabilistic analogue of NP. Specifically, we construct a quantum version of the $k$-SAT problem which we call “stoquastic $k$-SAT” such that stoquastic $k$-SAT is contained in MA for any constant $k$, and any promise problem in MA is Karp-reducible to stoquastic 6-SAT. This result provides the first nontrivial example of a MA-complete promise problem.

Complex Quadratic Optimization and Semidefinite Programming

In this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the Hermitian complex form. A special case of the problem that we study corresponds to the max-3-cut model used in a recent paper of Goemans and Williamson J. Comput. System Sci., 68 (2004), pp. 442-470]. We first develop a closed-form formula to compute the probability of a complex-valued normally distributed bivariate random vector to be in a given angular region. This formula allows us to compute the expected value of a randomized (with a specific rounding rule) solution based on the optimal solution of the complex semidefinite programming relaxation problem. In particular, we present an $[m^2(1-\cos\frac{2\pi}{m})/8\pi]$-approximation algorithm, and then study the limit of that model, in which the problem remains NP-hard. We show that if the objective is to maximize a positive semidefinite Hermitian form, then the randomization-rounding procedure guarantees a worst-case performance ratio of $\pi/4 \approx 0.7854$, which is better than the ratio of $2/\pi \approx 0.6366$ for its counterpart in the real case due to Nesterov. Furthermore, if the objective matrix is real-valued positive semidefinite with nonpositive off-diagonal elements, then the performance ratio improves to 0.9349.

Laplace Spectra of Orgraphs and Their Applications

The Laplace matrix is the matrix L = (?ij) ? ? n × n with nonpositive off-diagonal elements and zero row sums. A weighted orgraph corresponds to each Laplace matrix, its properties being closely related to the algebraic properties of the Laplace matrix. The normalized Laplace matrix 719-01 is the Laplace matrix where ?1/n ? ?ij ? 0 for all i ? j. The paper was devoted to the spectrum of the Laplace matrices and to the relationship between the spectra of the Laplace and stochastic matrices. The normalized Laplace matrices were proved to be semiconvergent. It was established that the multiplicity of the eigenvalue 0 of the matrix $$\tilde L$$ is equal to the in-forest dimension of the corresponding orgraph, and the multiplicity of the eigenvalue 1 is one less than the in-forest dimension of the complementary orgraph. The spectra of the matrices $$\tilde L$$ belong to the intersection of two circles of radius 1 ? 1/n centered at the points 1/n and 1 ? 1/n, respectively. Additionally, the domain that comprises them is included in the intersection of two angles (defined in the paper) with vertices 0 and 1 and the band $$|\operatorname{Im} (z)|\; \leqslant \tfrac{1}{{2n}}\cot \tfrac{\pi }{{2n}}$$ (at the limit |Im(z)| < 1/?). A polygon with all points being the eigenvalues of the normalized n-order Laplace matrices was constructed.

Some subclasses of Petri nets and the analysis of their structural properties: a new approach

The purpose of the paper is to consider some special types of Petri nets, introduced by Lien (1976), and to propose a complete and unified approach for the study of their structural properties by using techniques of linear algebra of matrices. We distinguish four subclasses: forward-conflict-free, backward-conflict-free, forward-concurrent-free, and backward-concurrent-free Petri nets. A modification of the classical incidence matrix results in a square matrix, called a modified incidence matrix, with nonpositive (nonnegative) off-diagonal elements when backward-(forward-) conflict-free or concurrent-free Petri nets are considered. The modified incidence matrix eigenvalues are computed and theorems on matrices of this type are used to prove several sufficient and/or necessary conditions for structural boundedness, liveness, repetitiveness, conservativeness, and consistency of these four subclasses of Petri nets.

Structural Properties of a Class of Petri Nets

This paper introduces an extended version of Weighted Marked Graphs, called Extended Weighted Marked Graphs (EWMGs for short) in which each place has exactly one input transition. They are capable of naturally representing synchronization, concurrency and choice, and are suitable for modeling systems with bulk consumptions or productions of resources. A modification of the classical incidence matrix results on a square matrix with nonpositive off-diagonal elements. Theorems on matrices of this type are used, in this paper, to prove structural boundedness, structural liveness, conservativeness, consistency, repetitiveness, and well-structuredness of EWMGs. The application of our approach to an assembly manufacturing system shows how these results are easily computed compared to the existing methods for deciding these properties for a Petri net.

On the Jacobian Conjecture: Reduction of Coefficients

We prove that a polynomial map from Rn to itself with non-zero constant Jacobian determinant is a stably tame automorphism if its linear part is the identity and all the coefficients of its higher order terms are non-positive. We also prove that the Jacobian conjecture holds for any number of variables and any field of characteristic zero, if one can show that every polynomial map of Rn to itself is injective when it has a non-zero constant Jacobian determinant and has linear part the identity, and all the coefficients of higher order terms are non-negative. The proofs use special properties of matrices with non-positive off-diagonal elements and non-negative principal minors, and of matrices with vanishing principal minors. Furthermore we reduce the Jacobian conjecture to a polynomial matrix problem. Moreover, if the matrix has a positive answer, then every real polynomial automorphism is stably tame.

TheL-dominance concept and its applications to multivariable system designs

A new type of dominance measure based on triangular factorizations is developed and is named the L-dominance measure. This new measure is tighter than any existing dominance measures and in particular is tighter than those based on generalized dominance and M-matrix theory. The L-dominance condition on matrices with non-positive off-diagonal elements is proved to be equivalent to the M-matrix condition. But in contrast to the M-matrix condition, the testing of the L-dominance condition is easier and can be carried out in sequential order. Furthermore, these special properties enable us to construct a synthesis based design procedure superior to those based on generalized dominance or M-matrix theory; the last two methods are generally regarded as post-mortem analytical tools only.

New dominance concepts for multivariable control systems design

This paper sets out a revised view of the classical diagonal dominance method for multivariable feedback systems. The various types of dominance measure are unified by using one fundamental dominance theorem (FD)l. It is shown that the fundamental dominance condition yields the smallest dominance measure when compared with all the known classical diagonal dominance measures. In addition it is shown that the dominance measures such as row/column, generalized dominance and a few new ones are special cases and can be derived from this same FD theorem. This improved dominance criterion also provides the sufficient condition for a system to be approximated by a nominal one in the sense that the stabilization of this nominal system will assure the stability of the original one. Finally, a new type of dominance measure named L-dominance is presented. It has been shown that this new approach yields an even smaller dominance measure, and that when it is applied to matrices with non-positive off-diagonal elements, ...

A polynomial-time test for M-matrices

A real, square matrix with nonpositive off-diagonal elements is a Z-matrix, and a Z-matrix with nonnegative principal minors is an M-matrix. Using principal pivoting, we derive a polynomial-time algorithm for testing whether a Z-matrix is an M-matrix. Our algorithm is much simpler than a recent one proposed by Ramamurthy. We also study the zero-nonzero structure of the inverse of a nonsingular M-matrix and compare A -1 and B -1 in the case that A and B are nonsingular M-matrices and B ≧ A.

A bard-type method for a generalized linear complementarity problem with a nonsingular m-matrix

In this article we develop an extension of Murty's Bard-type method for the solution of a generalized linear complementarity problem with upper bounds (BLCP) when its matrix M has positive principal minors (M ∈ P). We prove that the Bard-type algorithm converges to the unique solution of the BLCP when M is a P-matrix with nonpositive off-diagonal elements. We also study two special cases of the BLCP and prove that the Bard-type method is convergent to the unique solution of these BLCPs when M ∈ P. We show that if M ∈ NSM in these two cases, then the efficiency of the algorithm can be improved, by exploiting some properties of these problems. Computational experience with the Bard-type algorithm in the solution of large-scale BLCPs with sparse NSM maytrices is also included and shows the efficiency of this approach.

A polynomial algorithm for testing the nonnegativity of principal minors of Z-matrices

A Z-matrix is a square matrix with nonpositive off-diagonal elements. We give a polynomial algorithm for testing the nonnegativity of principal minors of Z-matrices.

Matrices With Applications in Statistics

Prerequisite matrix theory. Prerequisite vector theory. Linear transformations and characteristic roots. Geometric interpretations. Algebra of vector spaces. Generalized inverse. Conditional inverse. Systems of linear equations. Patterned matrices and certain other special matrices. Trace of a matrix, vector of a matrix, commutation matrices. Integration and differentiation. Positive matrices and matrices with non-positive off-diagonal elements. Non-negative matrices, idempotent and tripotent matrices, projections.

On a generalized Fan inequality

It is shown that the ω- and τ-matrices, the weakly sign symmetric matrices, the R - and V -matrices, and the matrices c -equivalent to an M -matrix or to a real matrix with nonpositive off-diagonal elements, can all be characterized by the same determinantal inequality, which we call a generalized Fan inequality.