Related papers: A Combinatorial Result on Block Matrices
The complete eigenstructure, or structural data, of a rational matrix $R(s)$ is comprised by its invariant rational functions, both finite and at infinity, which in turn determine its finite and infinite pole and zero structures,…
We prove necessary and sufficient conditions on a family of (generalised) gridding matrices to determine when the corresponding permutation classes are partially well-ordered. One direction requires an application of Higman's Theorem and…
In this paper we give a novel solution to a classical completion problem for square matrices. This problem was studied by many authors through time, and it is completely solved in [2, 3]. In this paper we relate this classical problem to a…
Integer partitions are one of the most fundamental objects of combinatorics (and number theory), and so is enumerating objects avoiding patterns. In the present paper we describe two approaches for the systematic counting of classes of…
Efficient resource allocation is one of the main driving forces of human civilizations. Of the many existing approaches to resource allocation, matrix completion is one that is frequently applied. In this paper, we investigate a special…
A bistochastic matrix is a square matrix with positive entries such that rows and columns sum to unity. A unistochastic matrix is a bistochastic matrix whose matrix elements are the absolute values squared of a unitary matrix. We can now…
In this paper, we present sufficient conditions to guarantee the invertibility of rational circulant matrices with any given size. These sufficient conditions consist of linear combinations of the entries in the first row with integer…
Algorithms to generate various combinatorial structures find tremendous importance in computer science. In this paper, we begin by reviewing an algorithm proposed by Rohl that generates all unique permutations of a list of elements which…
It is well known that any nonsingular M-matrix admits an LU factorization into M-matrices (with L and U lower and upper triangular respectively) and any singular M-matrix is permutation similar to an M-matrix which admits an LU…
Criterion for a companion matrix to have a certain number of flat portions on the boundary of its numerical range is given. The criterion is specialized to the cases of 3-by-3 and 4-by-4 matrices. In the latter case, it is proved that a…
Vacillating tableaux are sequences of integer partitions that satisfy specific conditions. The concept of vacillating tableaux stems from the representation theory of the partition algebra and the combinatorial theory of crossings and…
The {\em line sum optimization problem} asks for a $(0,1)$-matrix minimizing the sum of given functions evaluated at its row and column sums. We show that the {\em uniform} problem, with identical row functions and identical column…
We give the first algorithm for Matrix Completion whose running time and sample complexity is polynomial in the rank of the unknown target matrix, linear in the dimension of the matrix, and logarithmic in the condition number of the matrix.…
To apportion a complex matrix means to apply a similarity so that all entries of the resulting matrix have the same magnitude. We initiate the study of apportionment, both by unitary matrix similarity and general matrix similarity. There…
In this paper, we consider matrix completion from non-uniformly sampled entries including fully observed and partially observed columns. Specifically, we assume that a small number of columns are randomly selected and fully observed, and…
Symmetry is an important problem in many combinatorial problems. One way of dealing with symmetry is to add constraints that eliminate symmetric solutions. We survey recent results in this area, focusing especially on two common and useful…
An $m \times (n+1)$ multiplicity matrix is a matrix $M = ( \mu_{i,j} )$ with rows enumerated by $i \in \{ 1,\ 2, \ldots, m \}$ and columns enumerated by $j \in \{ 0,1,\ldots, n \}$ whose coordinates are nonnegative integers satisfying the…
Cramer's rules for some left, right and two-sided quaternion matrix equations are obtained within the framework of the theory of the column and row determinants.
The paper studies the question of existence of polynomials with given roots over associative non-commutative rings with identity. It is shown that in the case of an associative division ring for arbitrary n elements of this ring there…
This article gives necessary and sufficient conditions for a relation to be the containment relation between the facets and vertices of a polytope. Also given here, are a set of matrices parameterizing the linear moduli space and another…