Related papers: Some combinatorial matrices and their LU-decomposi…
A combinatorial rectangle may be viewed as a matrix whose entries are all +-1. The discrepancy of an m by n matrix is the maximum among the absolute values of its m row sums and n column sums. In this paper, we investigate combinatorial…
We introduce combinatorial objects named matricubes that provide a generalization of the theory of matroids. As matroids provide a combinatorial axiomatization of hyperplane arrangements, matricubes provide a combinatorial axiomatization of…
In this paper we consider the decomposition of positive semidefinite matrices as a sum of rank one matrices. We introduce and investigate the properties of various measures of optimality of such decompositions. For some classes of positive…
We give a simple topological construction of the Burau representations of the loop braid groups. There are four versions: defined either on the non-extended or extended loop braid groups, and in each case there is an unreduced and a reduced…
We define and prove isomorphisms between three combinatorial classes involving labeled trees. We also give an alternative proof by means of generating functions.
We give necessary and sufficient conditions for the Zhang-Liu matrices to be diagonalizable over arbitrary fields and provide the eigen-decomposition when it is possible. We use this result to calculate the order of these matrices over any…
Matrices over the dual numbers are considered. We propose an approach to classify these matrices up to similarity. Some preliminary results on the realization of this approach are obtained. In particular, we produce explicitly canonical…
The matrix LU factorization algorithm is a fundamental algorithm in linear algebra. We propose a generalization of the LU and LEU algorithms to accommodate the case of a commutative domain and its field of quotients. This algorithm…
A simple matrix formulation of the Fibonacci, Lucas, Chebyshev, and Dixon polynomials polynomials is presented. It utilizes the powers and the symmetric tensor powers of a certain matrix.
Conjugation covariants of matrices are applied to study the real algebraic variety consisting of complex Hermitian matrices with a bounded number of distinct eigenvalues. A minimal generating system of the vanishing ideal of degenerate…
Multiple harmonic-like numbers are studied using the generating function approach. A closed form is stated for binomial sums involving these numbers and two additional parameters. Several corollaries and examples are presented which are…
{Let ${\Cal X}$ be a self-dual P-polynomial association scheme. Then there are at most 12 diagonal matrices $T$ such that $(PT)^3=I$. Moreover, all of the solutions for the classical infinite families of such schemes (including the Hamming…
A general theorem on factorization of matrices with polynomial entries is proven and it is used to reduce polynomial Darboux matrices to linear ones. Some new examples of linear Darboux matrices are discussed.
In this article we give a classification of the sub-groups in PSL(2,Z) and of the conjugacy classes of these sub-groups by the mean of an combinatorial invariant: some trivalent diagrams (dotted or not). We give explicit formulae enabling…
In this note we focus on combinatorial aspects of plus-one generated line arrangements. We provide combinatorial constraints on such arrangements and we construct a polynomial that decodes the plus-one generated property. We present new…
A novel factorization for the sum of two single-pair matrices is established as product of lower-triangular, tridiagonal, and upper-triangular matrices, leading to semi-closed-form formulas for tridiagonal matrix inversion. Subsequent…
Branes and defects in topological Landau-Ginzburg models are described by matrix factorisations. We revisit the problem of deforming them and discuss various deformation methods as well as their relations. We have implemented these…
The idea of decomposing a matrix into a product of structured matrices such as triangular, orthogonal, diagonal matrices is a milestone of numerical computations. In this paper, we describe six new classes of matrix decompositions,…
We study several variants of decomposing a symmetric matrix into a sum of a low-rank positive semidefinite matrix and a diagonal matrix. Such decompositions have applications in factor analysis and they have been studied for many decades.…
We settle the existence of certain "anti-magic" cubes using combinatorial block designs and graph decompositions to align a handful of small examples.