Related papers: Construction, classification and parametrization o…
A novel method to obtain parametrizations of complex inverse orthogonal matrices is provided. These matrices are natural generalizations of complex Hadamard matrices which depend on non zero complex parameters. The method we use is via…
We introduce Hadamard matrices whose entries are quaternionic. We then go on to provide classification of quaternionic Hadamard matrices of circulant core of orders 2 through 5. We also introduce quaternionic Hadamard matrices of Butson…
In this paper we use a design theoretical approach to construct new, previously unknown complex Hadamard matrices. Our methods generalize and extend the earlier results of de la Harpe--Jones and Munemasa--Watatani and offer a theoretical…
Complex Hadamard matrices H of order 6 are characterized in a novel manner, according to the presence/absence of order 2 Hadamard submatrices. It is shown that if there exists one such submatrix, H is equivalent to a Hadamard matrix where…
Graphs are very important mathematical structures used in many applications, one of which is transportation science. When dealing with transportation networks, one deals not only with the network structure, but also with information related…
Random matrices now play a role in many parts of computational mathematics. To advance these applications, it is desirable to have tools that are flexible, easy to use, and powerful. Over the last 25 years, researchers have developed a…
Matrices are very popular and widely used in mathematics and other fields of science. Every mathematician has known the properties of finite-sized matrices since the time of study. In this paper, we consider the basic theory of infnite…
The aim of this text is to provide a linguistically accessible, but comprehensive introduction into a variety of topics in dynamical systems and its applications. Whilst preliminary knowledge of dynamical systems is useful, it is not…
The defect of a complex Hadamard matrix $H$ is an upper bound for the dimension of a continuous Hadamard orbit stemming from $H$. We provide a new interpretation of the defect as the dimension of the center subspace of a gradient flow and…
In this article algebraic constructions are introduced in order to study the variety defined by a radical parametrization (a tuple of functions involving complex numbers, $n$ variables, the four field operations and radical extractions). We…
A complex Hadamard matrix is a square matrix H with complex entries of absolute value 1 satisfying $HH^*= nI$, where $*$ stands for the Hermitian transpose and I is the identity matrix of order $n$. In this paper, we first determine the…
In recent years, random matrices have come to play a major role in computational mathematics, but most of the classical areas of random matrix theory remain the province of experts. Over the last decade, with the advent of matrix…
In this chapter, I review the main methods and techniques of complex systems science. As a first step, I distinguish among the broad patterns which recur across complex systems, the topics complex systems science commonly studies, the tools…
A new class of structured matrices is presented and a closed form formula for their determinant is established. This formula has strong connections with the one for Vandermonde matrices.
We describe combinatorial properties of the defining row of a circulant Hadamard matrix by exploiting its orthogonality to subsequent rows, and show how to exclude several particular forms of these matrices.
We present in this paper some fundamental tools for developing matrix analysis over the complex quaternion algebra. As applications, we consider generalized inverses, eigenvalues and eigenvectors, similarity, determinants of complex…
Hadamard matrices are square $n\times n$ matrices whose entries are ones and minus ones and whose rows are orthogonal to each other with respect to the standard scalar product in $\Bbb R^n$. Each Hadamard matrix can be transformed to a…
Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence in terms of relatively simple invariants. Where…
The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research.…
We investigate an approach to matroid complexity that involves describing a matroid via a list of independent sets, bases, circuits, or some other family of subsets of the ground set. The computational complexity of algorithmic problems…