Nonnegative matrices having nonnegative Drazin pseudoinverses

Necessary and sufficient conditions for nonnegative matrices having nonnegative Drazin pseudoinverses are obtained. A decomposition theorem which characterizes the class of all nonnegative matrices with nonnegative Drazin pseudoinverses is proved, thus answering a question raised by several people. It is also shown that if a row (or column) stochastic matrix has a nonnegative Drazin pseudoinverse A(^d), then A(^d) is some power of A. These results extend known results for nonnegative group-monotone matrices.     

A geometric treatment of generalized inverses and semigroups of nonnegative matrices

The purpose of this paper is to provide a unified treatment from the geometric viewpoint of the following closely related aspects of nonnegative matrices: nonnegative matrices with nonnegative generalized inverses of various kinds; nonnegative rank factorization; regular elements, Green's relations, and maximal subgroups of the semigroups of nonnegative matrices, stochastic matrices, column stochastic matrices, and doubly stochastic matrices.     

Automorphisms of the semigroup of nonnegative invertible matrices of order two over partially ordered commutative rings

In the paper, the automorphisms of the semigroup of nonnegative invertible matrices of order two over a partially ordered commutative ring with 2 invertible are described.     

Stochastic matrices over cones

The notion of a stochastic operator in an ordered Banach space is specialized to a finite dimensional ordered real vector space. The classical limit theorems are obtained, and an application is made to non-homogeneous Markov chains. Finally, groups of nonnegative matrices are discussed.     

Stochastic matrices over cones

The notion of a stochastic operator in an ordered Banach space is specialized to a finite dimensional ordered real vector space. The classical limit theorems are obtained, and an application is made to non-homogeneous Markov chains. Finally, groups of nonnegative matrices are discussed.     

Row Stochastic Matrices Similar to Doubly Stochastic Matrices

The problem of determining which row stochastic n-by-n matrices are similar to doubly stochastic matrices is considered. That not all are is indicated by example, and an abstract characterization as well as various explicit sufficient conditions are given. For example, if a row stochastic matrix has no entry smaller than (n+1)^-1 it is similar to a doubly stochastic matrix.

Relaxing the nonnegativity requirement, the real matrices which are similar to real matrices with row and column sums one are then characterized, and it is observed that all row stochastic matrices have this property. Some remarks are then made on the nonnegative eigenvalue problem with respect to i) a necessary trace inequality and ii) removing zeroes from the spectrum.     

Error Bounds for Nonnegative Dynamic Models

The extension of Markov reward models to dynamic models with nonnegative matrices is motivated by practical applications, such as economic input–output, employment, or population models. This paper studies the generalization of error bound theorems for Markov reward structures to dynamic reward structures with arbitrary nonnegative matrices. Both irreducible and reducible matrices are covered. In addition, results for the stochastic case are unified and extended. First, generalized expressions are derived for average reward functions. The special normalization case is distinguished and is shown to be transformable into the stochastic case. Its interpretation is of interest in itself. Next, error bound results are studied. Under a general normalization condition, it is shown that the results for the stochastic case can be extended. Both the average case and the transient case are included. A random walk-type example is included to illustrate the results.     

Hadamard matrices of order 28 with automorphisms of order 7

The only primes which can divide the order of the automorphism group of a Hadamard matrix of order 28 are 13, 7, 3, and 2, and there are precisely four inequivalent matrices with automorphisms of order 13 (Tonchev, J. Combin. Theory Ser. A35 (1983), 43–57). In this paper we show that there are exactly twelve inequivalent Hadamard matrices of order 28 with automorphisms of order 7. In particular, there are precisely seven matrices with transitive automorphism groups.     

Nonnegative spectraloid matrices

Several characterizations are given for certain nonnegative matrices having equal spectral and numerical radii. Some of these results are applied to stochastic matrices.     

Comparison theorems for splittings of matrices

Summary. This paper investigates the comparisons of asymptotic rates of convergence of two iteration matrices. On the basis of nonnegative matrix theory, comparisons between two nonnegative splittings and between two parallel multisplitting methods are derived. When the coefficient matrix A is Hermitian positive (semi)definite, comparison theorems about two P-regular splittings and two parallel multisplitting methods are proved. Received April 4, 1998 / Revised version received October 18, 1999 / Published online November 15, 2001     

Measurability of automorphisms of topological groups

Relations between the measurability and continuity of algebraic automorphisms of topological groups depending on the types of groups are examined. Various cases are considered and theorems on the continuity of measurable automorphisms are proved; for instance, such theorems are proved for separable locally compact groups and automorphisms measurable with respect to nonnegative Haar measures. On the other hand, examples of nonmetrizable nonseparable compact groups with Haar measures and of non-locally-compact separable metrizable groups with measures μ quasi-invariant with respect to dense subgroups admittings μ-measurable discontinous automorphisms are given. Translated fromMatenmaticheskie Zametki, Vol. 68, No. 1, pp. 105–112, July, 2000. An erratum to this article can be found online at .     

Automorphisms of the semigroup of nonnegative invertible matrices of order 2 over rings

In this paper, we describe automorphisms of the semigroup G₂(R) of nonnegative invertible matrices if R is a (not necessarily commutative) partially ordered ring without zero divisors with 1/n for some natural number n?>?1.     

Generalization of some results concerning eigenvalues of a certain class of matrices and some applications

In this note, we present a generalization of some results concerning the spectral properties of a certain class of block matrices. As applications, we study some of its implications on nonnegative matrices and doubly stochastic matrices as well as on graph spectra and graph energy.     

Nonnegative tensors revisited: plane stochastic tensors

In this paper, we develop and enrich the theory of nonnegative tensors. We define the sign nonsingular tensors and establish the relationship between the combinatorial determinant and the permanent of nonnegative tensors. We generalize the results from doubly stochastic matrices to totally plane stochastic tensors and obtain a probabilistic algorithm for locating a positive diagonal in a nonnegative tensor under certain conditions. We form a normalization algorithm to convert some nonnegative tensors to plane stochastic tensors. We obtain a lower bound for the minimum of the axial N-index assignment problem by means of the set of plane stochastic tensors.     

On the genus of the groups PSL(2, q), PSL(3, q), and PSp(4, q)

SL(n, q) is the group of n×n matrices, over the Galois field GF(q), of determinate 1. PSL(n, q) is SL(n, q) modulo the scalar n×n matrices of determinate 1. PSL(n, q) acts on the Desarguesian projective space PG(n−1, q). Sp(4, q) is the group of 4 × 4 matrices of determinate 1 which preserve the symplectic bilinear form on the 4 × 1 matrices over GF(q). PSp(4, q) is Sp(4, q) modulo Z = {1,−1}. PSp(4, q) acts on the symplectic generalized quadrangle W(3, q), a subspace of the projective space PG(3, q), as a group of automorphisms. In this paper, bounds are given for the genus of these groups.     

Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves

The algebra of quantum matrices of a given size supports a rational torus action by automorphisms. It follows from work of Letzter and the first named author that to understand the prime and primitive spectra of this algebra, the first step is to understand the prime ideals that are invariant under the torus action. In this paper, we prove that a family of quantum minors is the set of all quantum minors that belong to a given torus-invariant prime ideal of a quantum matrix algebra if and only if the corresponding family of minors defines a non-empty totally nonnegative cell in the space of totally nonnegative real matrices of the appropriate size. As a corollary, we obtain explicit generating sets of quantum minors for the torus-invariant prime ideals of quantum matrices in the case where the quantisation parameter q is transcendental over ${\mathbb{Q}}$ .     

整系数射影么模羣的自同构

<正> §1.序言 以表行列式之值为±1的n×n整系数矩陣所組成的乘法羣,而以表中行列式之值为+1的矩陣所組成的子羣.的中核由{I,-I}所組成(I是单位矩阵).以表对其中核的商羣,称之为的射影羣.当n是偶数吋,的中核也是由{I,-I}所組成,以表对其中核之商羣,称之为整系数射影模羣. 华罗庚教授和I.Reiner在[1]中决定了,及(m≥1)的自同构.当n是奇     

2-(31,15,7), 2-(35,17,8) and 2-(36,15,6) designs with automorphisms of odd prime order,and their related Hadamard matrices and codes

We present the full classification of Hadamard 2-(31,15,7), Hadamard 2-(35, 17,8) and Menon 2-(36,15,6) designs with automorphisms of odd prime order. We also give partial classifications of such designs with automorphisms of order 2. These classifications lead to related Hadamard matrices and self-dual codes. We found 76166 Hadamard matrices of order 32 and 38332 Hadamard matrices of order 36, arising from the classified designs. Remarkably, all constructed Hadamard matrices of order 36 are Hadamard equivalent to a regular Hadamard matrix. From our constructed designs, we obtained 37352 doubly-even [72,36,12] codes, which are the best known self-dual codes of this length until now.