On the maximum of per(I-A)

The purpose of this paper is to give some necessary conditions on a substochastic matrix which maximizes the values of Per(I-A) for A taken in the semigroup of n × n substochastic matrices, and to determine the exact value of the maximum which is found to be 2^[n/2].     

Optimal stopping,exponential utility,and linear programming

This paper uses linear programming to compute an optimal policy for a stopping problem whose utility function is exponential. This is done by transforming the problem into an equivalent one having additive utility and nonnegative (not necessarily substochastic) transition matrices.Research was supported by NSF Grant ENG 76-15599.     

A minimal completion of (0, 1)-matrices without total support

In this paper, we provide a method to complete a (0, 1)-matrix without total support via the minimal doubly stochastic completion of doubly substochastic matrices and show that the size of the completion is determined by the maximum diagonal sum or the term rank of the given (0, 1)-matrix.     

Pseudomoments of the Riemann zeta-function and pseudomagic squares

We compute integral moments of partial sums of the Riemann zeta function on the critical line and obtain an expression for the leading coefficient as a product of the standard arithmetic factor and a geometric factor. The geometric factor is equal to the volume of the convex polytope of substochastic matrices and is equal to the leading coefficient in the expression for moments of truncated characteristic polynomial of a random unitary matrix.     

Transient policies in discrete dynamic programming: Linear programming including suboptimality tests and additional constraints

This paper investigates the computation of transient-optimal policies in discrete dynamic programming. The model, is quite general: it may contain transient as well as nontransient policies. and the transition matrices are not necessarily substochastic. A functional equation for the so-called transient-value-vector is derived and the concept of superharmonicity is introduced. This concept provides the linear program to compute the transientvalue-vector and a transient-optimal policy. We also discuss the elimination of suboptimal actions, the solution of problems with additional constraints, and the computation of an efficient policy for a multiple objective dynamic programming problem.     

Generalized inverses of substochastic matrices

This paper looks at the question of when a substochastic matrix has a substochastic generalized inverse. This question is answered for several generalized inverses, including semiinverses, the Moore–Penrose inverse, and the group inverse. Methods for constructing all such inverses are given.     

On infinite doubly substochastic matrices

Summary An infinite matrix is said to be doubly substochastic if it has nonnegative components and each row and each column sum is at most 1. Let x and y be two real sequences which converge to 0 or which are absolutely summable. This paper introduces necessary and sufficient conditions for existence of an infinite doubly substochastic matrix A such that x=Ay concerning partial order and convex hull for sequences.     

On three Krein extension problems and some generalizations

Three basic extension problems which were initiated by M. G. Krein are discussed and further developed. Connections with interpolation problems in the Carathéodory class are explained. Some tangential and bitangential versions are considered. Full characterizations of the classes of resolvent matrices for these problems are given and formulas for the resolvent matrices of left tangential problems are obtained using reproducing kernel Hilbert space methods.Dedicated to the memory of M. G. Krein, a beacon for us both.The authors wish to acknowledge the partial support of the Israel-Ukraine Exchange Program. D. Z. Arov also wishes to thank the Weizmann Institute of Science for partial support and hospitality; H. Dym wishes to thank Renee and Jay Weiss for endowing the chair which supports his research.     

On classes of matrices containing M-matrices and hermitian positive semidefinite matrices

Two new classes of matrices are introduced, containing hermitian positive semi-definite matrices and M-matrices. The relation to other well-known classes such as ω and τ-matrices and weakly sign symmetric matrices is examined, and invariance properties are shown.     

Classes of stable complex matrices defined via the theorems of Geršgorin and Lyapunov

We consider (and characterize) mainly classes of (positively) stable complex matrices defined via methods of Ger?gorin and Lyapunov. Although the real matrices in most of these classes have already been studied, we sometimes improve upon (and even correct) what has been previously published. Many of the classes turn out quite naturally to be the products of common sets of matrices. A Venn diagram shows how the classes are related.     

ALPS: Classes of stable and semipositive matrices

The purpose of this paper is to characterize and interrelate various degrees of stability and semipositivity for real square matrices. The standard conditions for three major classes of matrices are made both stronger and weaker, and the resulting classes are examined. The major classes are diagonally stable, stable, and semipositive matrices, denoted by $\text{A,L,}$ and $\text{S,}$ respectively. Their relationship to the classes of matrices whose principal minors are positive (denoted by $\text{P}$ and non-negative (denoted by $\text{P}$₀) is also presented.     

Monotone positive stable matrices

The class of real matrices which are both monotone (inverse positive) and positive stable is investigated. Such matrices, called N-matrices, have the well-known class of nonsingular M-matrices as a proper subset. Relationships between the classes of N-matrices, M-matrices, nonsingular totally nonnegative matrices, and oscillatory matrices are developed. Conditions are given for some classes of matrices, including tridiagonal and some Toeplitz matrices, to be N-matrices.     

Combinatorial properties of some classes of matrices over GF(2)

We study several classes of matrices of GF(2) constructed from lists of subsets of finite sets In this paper. We show that all matrices in these classes are representations of connected equicardinal matrix over GF(2). In Matrix terms, these irreducible (defined below) matrices all have the property that every minimal dependent set of column has the same cardinality over GF(2). This fact is shown directly in this paper by elementary matrix considerations. In a subsequent paper, we shall show that these classes of matrices are in fact the classes of canonical forms for all representations of nontrivial binary connected equicardinal matroids.     

在两循环矩阵类中求解线性方程组反问题的快速算法

本文利用多项式最大公因式 ,给出了线性方程组的反问题在 r-循环矩阵类和对称 r-循环矩阵类中有唯一解的充要条件 ,进而得到线性方程组在 r循环矩阵类和对称 r-循环矩阵类中的反问题求唯一解的算法 .最后给出了应用该算法的数值例子 .     

Sub-direct sums and differences for certain matrix classes

The classes of accretive, dissipative, and accretive-dissipative matrices are shown to be closed with respect to taking a sub-direct sum, while the classes of SQD and BG matrices are closed with respect to taking a sub-direct difference. Computational consequences of these facts are discussed.     

An explicit example of a maximal 3-cyclically monotone operator with bizarre properties

Subdifferential operators of proper convex lower semicontinuous functions and, more generally, maximal monotone operators are ubiquitous in optimization and nonsmooth analysis. In between these two classes of operators are the maximal n$n$-cyclically monotone operators. These operators were carefully studied by Asplund, who obtained a complete characterization within the class of positive semidefinite (not necessarily symmetric) matrices, and by Voisei, who presented extension theorems à la Minty.     

Weaving hadamard matrices with maximum excess and classes with small excess

Weaving is a matrix construction developed in 1990 for the purpose of obtaining new weighing matrices. Hadamard matrices obtained by weaving have the same orders as those obtained using the Kronecker product, but weaving affords greater control over the internal structure of matrices constructed, leading to many new Hadamard equivalence classes among these known orders. It is known that different classes of Hadamard matrices may have different maximum excess. We explain why those classes with smaller excess may be of interest, apply the method of weaving to explore this question, and obtain constructions for new Hadamard matrices with maximum excess in their respective classes. With this method, we are also able to construct Hadamard matrices of near‐maximal excess with ease, in orders too large for other by‐hand constructions to be of much value. We obtain new lower bounds for the maximum excess among Hadamard matrices in some orders by constructing candidates for the largest excess. For example, we construct a Hadamard matrix with excess 1408 in order 128, larger than all previously known values. We obtain classes of Hadamard matrices of order 96 with maximum excess 912 and 920, which demonstrates that the maximum excess for classes of that order may assume at least three different values. Since the excess of a woven Hadamard matrix is determined by the row sums of the matrices used to weave it, we also investigate the properties of row sums of Hadamard matrices and give lists of them in small orders. © 2004 Wiley Periodicals, Inc. J Combin Designs 12: 233–255, 2004.     

Eigenvalue perturbation theory of structured real matrices and their sign characteristics under generic structured rank-one perturbations

An eigenvalue perturbation theory under rank-one perturbations is developed for classes of real matrices that are symmetric with respect to a non-degenerate bilinear form, or Hamiltonian with respect to a non-degenerate skew-symmetric form. In contrast to the case of complex matrices, the sign characteristic is a crucial feature of matrices in these classes. The behaviour of the sign characteristic under generic rank-one perturbations is analyzed in each of these two classes of matrices. Partial results are presented, but some questions remain open. Applications include boundedness and robust boundedness for solutions of structured systems of linear differential equations with respect to general perturbations as well as with respect to structured rank perturbations of the coefficients.     

Saddle point theorems

This paper considers three classes of matrices in terms of the existence ofsaddle points. The classes are described by conditions which are very closely related to the property ofquasi-concavity-convexity of functions of two variables. For the matrices in those classes, necessary and sufficient conditions for the existence of saddle points have been found.     

Decompositions of some types of quaternionic matrices

The structures of some important types of matrices over the quaternion skew field are discussed, and corresponding decompositions are obtained by virtue of real orthogonal matrices and real anti-symmetric matrices. In particular, the normal forms are found for several classes of quaternionic matrices.