On a new class of productive regular Hadamard matrices

We show that for each integer n for which there is a Hadamard matrix of order 4n and 8n²-1 is a prime number, there is a productive regular Hadamard matrix of order 16n²(8n²-1)². As a corollary, by applying a recent result of Ionin, we get many parametrically new classes of symmetric designs whenever either of 4n(8n²-1)-1 or 4n(8n²-1)+1 is a prime power.     

Infinite partition regular matrices

We consider infinite matrices with entries from (and only finitely many nonzero entries on any row). A matrixA is partition regular over provided that, whenever the set of positive integers is partitioned into finitely many classes there is a vector with entries in such that all entries ofA lie in the same cell of the partition. We show that, in marked contrast with the situation for finite matrices, there exists a finite partition of no cell of which contains solutions for all partition regular matrices and determine which of our pairs of matrices must always have solutions in the same cell of a partition.     

Extending subpermutation matrices in regular classes of matrices

We determine precise conditions in order that every n × n matrix of 0's and 1's with exactly k 1's in each row and column has the property that each subpermutation matrix of rank d can be extended to a permutation matrix. An application is given to completing partial latin squares.     

Comparisons of regular splittings of matrices

Summary In this article, new comparison theorems for regular splittings of matrices are derived. In so doing, the initial results of Varga in 1960 on regular splittings of matrices, and the subsequent unpublished results of Wonicki in 1973 on regular splittings of matrices, will be seen to be special cases of these new comparison theorems.Dedicated to Fritz Bauer on the occasion of his 60th birthdayResearch supported in part by the Air Force Office of Scientific Research, and by the Department of Energy     

On a class of coherent regular rings

The paper investigates a special class of quasi-local rings. It is shown that these rings are coherent and regular in the sense that every finitely generated submodule of a free module has a finite free resolution.

     

Structure of a class of unimodular matrices

A decomposition theorem is established for square matrices A(s) defined over R[s], the ring of real polynomials in a variable s, which satisfy the condition A(s)A(?s)=I.     

Sign regular matrices and Neville elimination

A pivoting strategy of O(n) operations for the Neville elimination of n × n nonsingular sign regular matrices is introduced. Among other nice properties, it is proved that it preserves sign regularity. It is also shown its relationship with scaled partial pivoting strategies for Neville elimination.     

Convergent regular splittings for nonnegative matrices

MP matrices are those real matrices which possess a nonnegative, nonsingular l-inverse. This paper characterizes the nonnegative MP matrices and hence, determines when a nonnegative matrix A has a convergent regular splitting M—Q which induces the linear stationary iterative scheme x_k+1=M^-1Qx_k+M^-1b to solve Ax=b.     

On a class of divisors of matrices

We propose a structural criterion for divisibility of matrices over a commutative principal ideal domain. Under certain restrictions we exhibit a method of finding all the non-associate divisors that have a prescribed canonical diagonal form.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 13–19.     

Nonsingular almost strictly sign regular matrices

The class of nonsingular almost strictly totally positive matrices has been characterized [M. Gasca, J.M. Peña, Characterizations and decompositions of almost strictly positive matrices, SIAM J. Matrix Anal. Appl. 28 (2006) 1–8]. In this paper, we discuss the class of almost strictly sign regular matrices that includes almost strictly totally positive matrices. A characterization is provided for these matrices in terms of their nontrivial minors using consecutive rows and consecutive columns. In particular, we present a characterization of certain almost strictly sign regular matrices in terms of a very reduced number of boundary almost trivial minors.     

Scattering matrices of regular coverings of graphs

We give a decomposition formula for the determinant on the bond scattering matrix of a regular covering of G. Furthermore, we define an L-function of G, and give a determinant expression of it. As a corollary, we express the determinant on the bond scattering matrix of a regular covering of G by means of its L-functions.