Some properties of the Lozinskii logarithmic norm

We study properties of the logarithmic matrix norm. We obtain new estimates for matrix norms as well as for the spectral radius and the spectral abscissa. We give a new proof of the Fiedler theorem and a block test for the Hurwitz property of a matrix based on the theory of nonnegative and off-diagonal-nonnegative matrices.     

非负矩阵最大特征值的新界值

对最大特征值的上下界进行估计是非负矩阵理论的重要部分,借助两个新的矩阵,从而得到一个判定非负矩阵最大特征值范围的界值定理,其结果比有关结论更加精确.     

Intervals of Totally Nonnegative and Related Matrices

We consider the class of the totally nonnegative matrices, i.e., the matrices having all their minors nonnegative, and intervals of matrices with respect to the chequerboard partial ordering, which results from the usual entrywise partial ordering if we reverse the inequality sign in all components having odd index sum. For these intervals we study the following conjecture: If the left and right endpoints of an interval are nonsingular and totally nonnegative then all matrices taken from the interval are nonsingular and totally nonnegative. We present a new class of the totally nonnegative matrices for which this conjecture holds true. Similar results for classes of related matrices are also given.     

On the geometric interpretation of the nonnegative rank

The nonnegative rank of a nonnegative matrix is the minimum number of nonnegative rank-one factors needed to reconstruct it exactly. The problem of determining this rank and computing the corresponding nonnegative factors is difficult; however it has many potential applications, e.g., in data mining and graph theory. In particular, it can be used to characterize the minimal size of any extended reformulation of a given polytope. In this paper, we introduce and study a related quantity, called the restricted nonnegative rank. We show that computing this quantity is equivalent to a problem in computational geometry, and fully characterize its computational complexity. This in turn sheds new light on the nonnegative rank problem, and in particular allows us to provide new improved lower bounds based on its geometric interpretation. We apply these results to slack matrices and linear Euclidean distance matrices and obtain counter-examples to two conjectures of Beasley and Laffey, namely we show that the nonnegative rank of linear Euclidean distance matrices is not necessarily equal to their dimension, and that the rank of a matrix is not always greater than the nonnegative rank of its square.     

Generalized eigenvectors and sets of nonnegative matrices

We present extensions of the Perron-Frobenius theory for square irreducible nonnegative matrices. After discussing structural properties of reducible nonnegative matrices we extend the theory to sets of nonnegative matrices, which play an important role in several dynamic programming recursions (e.g. Markov decision processes) and in mathematical economics (e.g. Leontief substitution systems). A set $\text{K}$ of (in general, reducible) matrices is considered, which is generated by all possible interchanges of corresponding rows, selected from a fixed finite set of square nonnegative matrices. A simultaneous block-triangular decomposition of the set of matrices $\text{K}$ is presented and characterized in terms of the maximal spectral radius, the maximal index, and generalized eigenvectors. As a by-product of our analysis we obtain a generalization of Howard's policy iteration method.     

Triangular truncation and its extremal matrices

The triangular truncation operator is a linear transformation that maps a given matrix to its strictly lower triangular part. The operator norm (with respect to the matrix spectral norm) of the triangular truncation is known to have logarithmic dependence on the dimension, and such dependence is usually illustrated by a specific Toeplitz matrix. However, the precise value of this operator norm as well as on which matrices can it be attained is still unclear. In this article, we describe a simple way of constructing matrices whose strictly lower triangular part has logarithmically larger spectral norm. The construction also leads to a sharp estimate that is very close to the actual operator norm of the triangular truncation. This research is directly motivated by our studies on the convergence theory of the Kaczmarz type method (or equivalently, the Gauß‐Seidel type method), the corresponding application of which is also included. Copyright © 2016 John Wiley & Sons, Ltd.     

Parallel multisplitting two-stage iterative methods with general weighting matrices for non-symmetric positive definite systems

In this work, we propose a new parallel multisplitting iterative method for non-symmetric positive definite linear systems. Based on optimization theory, the new method has two great improvements; one is that only one splitting needs to be convergent, and the other is that the weighting matrices are not scalar and nonnegative matrices. The convergence of the new parallel multisplitting iterative method is discussed. Finally, the numerical results show that the new method is effective.     

New lower bound of the determinant for Hadamard product on some totally nonnegative matrices

Applying the properties of Hadamard core for totally nonnegative matrices, we give new lower bounds of the determinant for Hadamard product about matrices in Hadamard core and totally nonnegative matrices, the results improve Oppenheim inequality for tridiagonal oscillating matrices obtained by T. L. Markham.     

The logarithmic norm. History and modern theory

In his 1958 thesis Stability and Error Bounds, Germund Dahlquist introduced the logarithmic norm in order to derive error bounds in initial value problems, using differential inequalities that distinguished between forward and reverse time integration. Originally defined for matrices, the logarithmic norm can be extended to bounded linear operators, but the extensions to nonlinear maps and unbounded operators have required a functional analytic redefinition of the concept.This compact survey is intended as an elementary, but broad and largely self-contained, introduction to the versatile and powerful modern theory. Its wealth of applications range from the stability theory of IVPs and BVPs, to the solvability of algebraic, nonlinear, operator, and functional equations. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65L05     

On invertible nonnegative Hamiltonian operator matrices

Some new characterizations of nonnegative Hamiltonian operator matrices are given. Several necessary and sufficient conditions for an unbounded nonnegative Hamiltonian operator to be invertible are obtained, so that the main results in the previously published papers are corollaries of the new theorems. Most of all we want to stress the method of proof. It is based on the connections between Pauli operator matrices and nonnegative Hamiltonian matrices.     

Complete controllability and contractibility in multimodal systems

The concept of a strictly positive definite set of Hermitian matrices is introduced. It is shown that a strictly positive definite set is always a positive definite set, and conditions are found under which a positive definite set is strictly positive definite. We also show that a set of Hermitian matrices is strictly positive definite if and only if some nonnegative linear combination of these matrices is a positive definite matrix. For state dimension two, we use this concept to find necessary and sufficient conditions for a two-mode completely controllable irreducible multimodal system to be contractible relative to an elliptic norm. For general state dimensions, we give necessary and sufficient conditions for a special-type two-mode completely controllable irreducible system to be contractible relative to a weakly monotone norm. Applying the above results, we show that, for state dimension two, there exists a completely controllable two-mode system which is not contractible relative to either an elliptic or a weakly monotone norm. We leave open the question whether or not complete controllability implies contractibility, relative to some norm, for multimodal systems of two or more modes.     

非负不可约矩阵Perron根新的下界序列

Estimate bounds for the Perron root of a nonnegative matrix are important in theory of nonnegative matrices. It is more practical when the bounds are expressed as an easily calculated function in elements of matrices. For the Perron root of nonnegative irreducible matrices, three sequences of lower bounds are presented by means of constructing shifted matrices, whose convergence is studied. The comparisons of the sequences with known ones are supplemented with a numerical example.     

非负矩阵Perron根的上界

文章针对特殊的非负矩阵，应月简单的相似变换，使矩阵保持非负性且最大行和减小，从而得到行和为正非负矩阵Perron根的新上界．     

Nonnegative Rank Factorization of a Nonnegative Matrix A with A A ≥0

In this article we obtain a nonnegative rank factorization of nonnegative matrices A satisfying one or both of the following conditions: (i) AA ? ? 0 (ii) A ? A ? 0, thus providing a new set of conditions that guarantee the existence of a nonnegative least-squares solution of a linear system. Indeed, the characterization of such matrices improves some of the previous known conditions for the existence of a nonnegative least-squares solution of a linear system.     

Nonnegative Rank Factorization of a Nonnegative Matrix A with A A ≥0

In this article we obtain a nonnegative rank factorization of nonnegative matrices A satisfying one or both of the following conditions: (i) AA † ≥0 (ii) A † A ≥0, thus providing a new set of conditions that guarantee the existence of a nonnegative least-squares solution of a linear system. Indeed, the characterization of such matrices improves some of the previous known conditions for the existence of a nonnegative least-squares solution of a linear system.     

On the monotonity of incomplete factorization

Summary The Meijerink, van der Vorst type incomplete decomposition uses a position set, where the factors must be zero, but their product may differ from the original matrix. The smaller this position set is, the more the product of incomplete factors resembles the original matrix. The aim of this paper is to discuss this type of monotonity. It is shown using the Perron Frobenius theory of nonnegative matrices, that the spectral radius of the iteration matrix is a monotone function of the position set. On the other hand no matrix norm of the iteration matrix depends monotonically on the position set. Comparison is made with the modified incomplete factorization technique.     

Merit functions for semi-definite complemetarity problems

Merit functions such as the gap function, the regularized gap function, the implicit Lagrangian, and the norm squared of the Fischer-Burmeister function have played an important role in the solution of complementarity problems defined over the cone of nonnegative real vectors. We study the extension of these merit functions to complementarity problems defined over the cone of block-diagonal symmetric positive semi-definite real matrices. The extension suggests new solution methods for the latter problems. This research is supported by National Science Foundation Grant CCR-9311621.     

Totally nonnegative cells and matrix Poisson varieties

We describe explicitly the admissible families of minors for the totally nonnegative cells of real matrices, that is, the families of minors that produce nonempty cells in the cell decompositions of spaces of totally nonnegative matrices introduced by A. Postnikov. In order to do this, we relate the totally nonnegative cells to torus orbits of symplectic leaves of the Poisson varieties of complex matrices. In particular, we describe the minors that vanish on a torus orbit of symplectic leaves, we prove that such families of minors are exactly the admissible families, and we show that the nonempty totally nonnegative cells are the intersections of the torus orbits of symplectic leaves with the spaces of totally nonnegative matrices.     

Stability of central finite difference schemes for the Heston PDE

This paper deals with stability in the numerical solution of the prominent Heston partial differential equation from mathematical finance. We study the well-known central second-order finite difference discretization, which leads to large semidiscrete systems with nonnormal matrices A. By employing the logarithmic spectral norm we prove practical, rigorous stability bounds. Our theoretical stability results are illustrated by ample numerical experiments. We also apply the analysis to obtain useful stability bounds for time discretization methods.     

非负矩阵Perron根的下界序列

非负矩阵Perron根的估计是非负矩阵理论研究的重要课题之一.如果其上下界能够表示为非负矩阵元素的易于计算的函数,那么这种估计价值更高.本文结合非负矩阵的迹分两种情况给出Perron根的下界序列,并且给出数值例子加以说明.