Related papers: A sign pattern that allows oppositely signed ortho…
Let T be a tile in the Cartesian plane made up of finitely many rectangles whose corners have rational coordinates and whose sides are parallel to the coordinate axes. This paper gives necessary and sufficient conditions for a square to be…
The rational invariants of the SL_2(q)-invariant quadratic forms on the real irreducible representations are determined. There is still one open question (see Remark 6.5) if q is an even square.
The q-classical orthogonal polynomials of the q-Hahn Tableau are characterized from their orthogonality condition and by a first and a second structure relation. Unfortunately, for the q-semiclassical orthogonal polynomials (a…
An alternating sign matrix is a square matrix satisfying (i) all entries are equal to 1, -1 or 0; (ii) every row and column has sum 1; (iii) in every row and column the non-zero entries alternate in sign. The 8-element group of symmetries…
We develop objective linear algebra in a new setting with a cardinality functor that can take negative values. The signs arise as little homotopies, as ratios between orientations. To illustrate the workings of the theory we give an…
We study a natural q-analogue of a class of matrices with noncommutative entries, which were first considered by Yu. I. Manin in 1988 in relation with quantum group theory, (called Manin Matrices in [5]) . These matrices we shall call…
A real matrix is said to be positive if its every entry is positive, and a real square matrix A is algebraically positive if there exists a real polynomial f such that f(A) is a positive matrix. A sign pattern matrix A is said to require a…
The combined matrix is a very useful concept for many applications. Almost strictly sign regular (ASSR) matrices form an important structured class of matrices with two possible zero patterns, which are either type-I staircase or type-II…
A new class of sign-symmetric matrices is introduced in this paper. Such matrices are named J--sign-symmetric. The spectrum of a J--sign-symmetric irreducible matrix is studied under assumptions that its second compound matrix is also…
The aim of this paper is to derive (by using two operators, representable by a Jacobi matrix) a family of q-orthogonal polynomials, which turn to be dual to alternative q-Charlier polynomials. A discrete orthogonality relation and a…
The pattern of a matrix M is a (0,1)-matrix which replaces all non-zero entries of M with a 1. A directed graph is said to support M if its adjacency matrix is the pattern of M. If M is an orthogonal matrix, then a digraph which supports M…
A $n\times n$ matrix $A$, which has a certain sign-symmetric structure ($J$--sign-symmetric), is studied in this paper. It is shown that such a matrix is similar to a nonnegative matrix. The existence of the second in modulus positive…
Let $(P_n)_n$ and $(Q_n)_n$ be two sequences of monic polynomials linked by a type structure relation such as $$ Q_{n}(x)+r_nQ_{n-1}(x)=P_{n}(x)+s_nP_{n-1}(x)+t_nP_{n-2}(x)\;, $$ where $(r_n)_n$, $(s_n)_n$ and $(t_n)_n$ are sequences of…
A banded matrix is a real square matrix where nonzero entries appear around the main diagonal. In this article, we consider linear complementarity properties of (variants) of banded matrices. Focusing on triangular matrices and the newly…
We use a greedy probabilistic method to prove that for every $\epsilon > 0$, every $m\times n$ Latin rectangle on $n$ symbols has an orthogonal mate, where $m=(1-\epsilon)n$. That is, we show the existence of a second Latin rectangle such…
The ensemble of antagonistic matrices is introduced and studied. In antagonistic matrices the entries $\mathcal A_{i,j}$ and $\mathcal A_{j,i}$ are real and have opposite signs, or are both zero, and the diagonal is zero. This…
A sign pattern is a matrix whose entries are from the set $\{+,-, 0\}$. A square sign pattern $A$ is called sign $k$-potent if $k$ is the smallest positive integer for which $A^{k+1}=A$, and for $k=1$, $A$ is called sign idempotent. In…
We study a one-parameter family of vector-valued polynomials associated to each simple Lie algebra. When this parameter $q$ equals -1 one recovers Joseph polynomials, whereas at $q$ cubic root of unity one obtains ground state eigenvectors…
We introduce a notion of compatibility for multiplicity matrices. This gives rise to a necessary condition for the join of two (possibly disconnected) graphs $G$ and $H$ to be the pattern of an orthogonal symmetric matrix, or equivalently,…
In this paper we study sequences of vector orthogonal polynomials. The vector orthogonality presented here provides a reinterpretation of what is known in the literature as matrix orthogonality. These systems of orthogonal polynomials…