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...

Let G be a simple undirected graph on n vertices and let S(G) be the class of real symmetric n× n matrices whose nonzero off-diagonal entries correspond exactly to the edges of G. Given 2n − 1 real numbers λ1 ≥ μ1 ≥ λ2 ≥ μ2 ≥ · · · ≥ λn−1 ≥ μn−1 ≥ λn, and a vertex v of G, the question is addressed of whether or not there exists A ∈ S(G) with eigenvalues λ1, . . . , λn such that A(v) has eigenva...

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, ...

the cyclic products of matrices and diagonal similarity. In this paper we consider diagonal similarity for matrices, which may be infinite, and whose elements lie in a (possible non-commutative) group G with O. Let H be a subgroup of a group G and let A be an irreducible square matrix with entries in GO. In Theorem 3.4, we give necessary and sufficient conditions for the existence of a matrix B...

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...

2 This paper considers the subexponential asymptotics of the stationary distributions of GI/G/1-type Markov chains in two cases: (i) the phase transition matrix in non-boundary levels is stochastic; and (ii) it is strictly substochastic. For the case (i), we present a weaker sufficient condition for the subexponential asymptotics than those given in the literature. As for the case (ii), the sub...

Cities have long served as nucleating centers for human development and advancement. facilitated the spread of both creativity disease, at same time, efforts to minimize disease influenced design cities. The purpose this paper is explore dynamics epidemics on networks in order help a multi-network city future aimed minimizing epidemics. In do this, we start with SIR model (susceptible, infected...

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....