We study the structure of 01-matrices avoiding a pattern P as an interval minor. We focus on critical P -avoiders, i.e., on the P -avoiding matrices in which changing a 0-entry to a 1-entry always creates a copy of P as an interval minor. Let Q be the 3 × 3 permutation matrix corresponding to the permutation 231. As our main result, we show that for every pattern P that has no rotated copy of Q...

Intertwiners between A–D–E lattice models are presented and the general theory developed. The intertwiners are discussed at three levels: at the level of the adjacency matrices, at the level of the cell calculus intertwining the face algebras and at the level of the row transfer matrices. A convenient graphical representation of the intertwining cells is introduced. The utility of the intertwin...

We show that error and erasure decoding of `-Interleaved Gabidulin codes, as well as list-` decoding of Mahdavifar–Vardy codes can be solved by row reducing skew polynomial matrices. Inspired by row reduction of F[x] matrices, we develop a general and flexible approach of transforming matrices over skew polynomial rings into certain normal forms. We apply this to solve generalised shift registe...

In this work an efficient algorithm to perform a block decomposition for large dense rectangular matrices with entries in 2 is presented. Matrices are stored as column blocks of row major matrices in order to facilitate rows operation and matrix multiplications with block of columns. One of the major bottlenecks of matrix decomposition is the pivoting involving both rows and column exchanges. S...

A 0; 1 matrix is balanced if, in every square submatrix with two nonzero entries per row and column, the sum of the entries is a multiple of four. This paper extends the decomposition of balanced 0; 1 matrices obtained by Conforti, Cornu ejols and Rao to the class of balanced 0; 1 matrices. As a consequence, we obtain a polynomial time algorithm for recognizing balanced 0; 1 matrices.

Abstract: Moments of secular and inverse secular coefficients, averaged over random matrices from classical groups, are related to the enumeration of non-negative matrices with prescribed row and column sums. Similar random matrix averages are related to certain configurations of vicious random walkers and to the enumeration of plane partitions. The combinatorial meaning of the average of the c...

When solving systems of equations by using matrices, many teachers present a Gauss-Jordan elimination approach to row reducing matrices that can involve painfully tedious operations with fractions (which I will call the traditional method). In this essay, I present an alternative method to row reduce matrices that does not introduce additional fractions until the very last steps. The students i...

We study properties of weight extraction methods for pairwise comparison matrices that minimize suitable measures of inconsistency, ‘average error gravity’measures, including one that leads to the geometric row means. The measures share essential global properties with the AHP inconsistency measure. By embedding the geometric mean in a larger class of methods we shed light on the choice between...

In this paper, regions containing eigenvalues of a matrix are obtained in terms of partial absolute deleted row sums and column sums. Furthermore, some sufficient and necessary conditions for H-matrices are derived. Finally, an upper bound for the Perron root of nonnegative matrices is presented. The comparison of the new upper bound with the known ones is supplemented with some examples.