16 research outputs found
Cohomology-Developed Matrices -- constructing families of weighing matrices and automorphism actions
The aim of this work is to construct families of weighing matrices via their
automorphism group action. This action is determined from the
-cohomology groups of the underlying abstract group. As a consequence,
some old and new families of weighing matrices are constructed. These include
the Paley Conference, the Projective-Space, the Grassmannian, and the
Flag-Variety weighing matrices. We develop a general theory relying on low
dimensional group-cohomology for constructing automorphism group actions, and
in turn obtain structured matrices that we call \emph{Cohomology-Developed
matrices}. This "Cohomology-Development" generalizes the Cocyclic and Group
Developments. The Algebraic structure of modules of Cohomology-Developed
matrices is discussed, and an orthogonality result is deduced. We also use this
algebraic structure to define the notion of \emph{Quasiproducts}, which is a
generalization of the Kronecker-product
Multilinear functional inequalities involving permanents, determinants, and other multilinear functions of nonnegative matrices and M-matrices
AbstractMotivated by a proof due to Fiedler of an inequality on the determinants of M-matrices and a recent paper by the authors, we now obtain various inequalities on permanents and determinants of nonsingular M-matrices. This is done by extending the multilinear considerations of Fiedler and, subsequently, of the authors, to fractional multilinear functionals on pairs of nonnegative matrices. Two examples of our results: For an n×n nonsingular M-matrix M (i) we give a sharp upper bound for det(M)+per(M), when M is a nonsingular M-matrix, (ii) we determine an upper bound on the relative error |per(M+E)−per(M)|/|per(M)|, when M+E is a certain componentwise perturbation of M
An improvement of an inequality of Fiedler leading to a new conjecture on nonnegative matrices
summary:Suppose that is an nonnegative matrix whose eigenvalues are . Fiedler and others have shown that , for all , with equality for any such if and only if is the simple cycle matrix. Let be the signed sum of the determinants of the principal submatrices of of order , . We use similar techniques to Fiedler to show that Fiedler’s inequality can be strengthened to: , for all . We use this inequality to derive the inequality that: . In the spirit of a celebrated conjecture due to Boyle-Handelman, this inequality inspires us to conjecture the following inequality on the nonzero eigenvalues of : If are (all) the nonzero eigenvalues of , then . We prove this conjecture for the case when the spectrum of is real
An algorithm for constructing and classifying the space of small integer weighing matrices
In this paper we describe an algorithm for generating all the possible
- integer Weighing matrices of weight up to
Hadamard equivalence. Our method is efficient on a personal computer for small
size matrices, up to , and . As a by product we also
improved the \textit{\textbf{nsoks}} \cite{riel2006nsoks} algorithm to find all
possible representations of an integer as a sum of integer squares.
We have implemented our algorithm in \texttt{Sagemath} and as an example we
provide a complete classification for \ and . Our list of
can serve as a step towards finding the open classical weighing
matrix