In this note I give a proof of the existence of a Smith normal form for matrices with integer entries. The existence of a good basis for a lattice with a finite index sublattice is a consequence of the Smith normal form. I conclude with an easy application to toric varieties. Theorem 1. Let A be a matrix with integer entries. Then there exist integer-values invertible matrices C and B such that...

Specially structured linear complementarity problems (LCPs) and their solution by the miss-cross method are examined. The criss-cross method is known to be finite for LCPs with positive semidefinite bisymmetric matrices and with P-matrices. It is also a simple finite algorithm for oriented matroid programming problems. Recently Cottle, Pang, and Venkateswaran identified the class of (column, ro...

A matrix with positive row sums and all its off-diagonal elements bounded above by their corresponding row averages is called a B-matrix by J. M. Peña in References (SIAM J. Matrix Anal. Appl. 2001; 22:1027–1037) and (Numer. Math. 2003; 95:337–345). In this paper, it is generalized to more extended matrices—MB-matrices, which is proved to be a subclass of the class of P-matrices. Subsequently, ...

Several classes of structured matrices (e.g., the Hadamard-Sylvester matrices and the pseudo-noise matrices) are used in the design of error-correcting codes. In particular, the columns of matrices belonging to the above two matrix classes are often used as codewords. In this paper we show that the two above classes essentially coincide: the pseudo-noise matrices can be obtained from the Hadama...

In this paper we obtain several results on the rank properties of Hadamard matrices (including Sylvester Hadamard matrices) as well as (generalized) Hadamard matrices. These results are used to show that the classes of (generalized) Sylvester Hadamard matrices and of generalized pseudo-noise matrices are equivalent, i.e., they can be obtained from each other by means of row/column permutations....

Two Hadamard matrices are considered equivalent if one is obtained from the other by a sequence of operations involving row or column permutations or negations. We complete the classification of Hadamard matrices of order 32. It turns out that there are exactly 13710027 such matrices up to equivalence. AMS Subject Classification: 05B20, 05B05, 05B30.

Some large rectangular matrices used in animal breeding are presented. We describe how to generate these matrices from the data supplied by animal breeders. The matrices are very sparse (3 nonzeros per row) and range between 26 20 and 968 652 582 694.

In this article a fast computational method is provided in order to calculate the Moore-Penrose inverse of full rank m× n matrices and of square matrices with at least one zero row or column. Sufficient conditions are also given for special type products of square matrices so that the reverse order law for the Moore-Penrose inverse is satisfied.

This paper presents a new technique for computing the barycenter of a set of distance or kernel matrices. These matrices, which define the interrelationships between points sampled from individual domains, are not required to have the same size or to be in row-by-row correspondence. We compare these matrices using the softassign criterion, which measures the minimum distortion induced by a prob...