Related papers: Algorithm for generating orthogonal matrices with …

This paper presents an algorithmic method for generating random orthogonal matrices \(A\) that satisfy the property \(A^t S A = S\), where \(S\) is a fixed real invertible symmetric or skew-symmetric matrix. This method is significant as it…

A \emph{rational orthogonal matrix} $Q$ is an orthogonal matrix with rational entries, and $Q$ is called \emph{regular} if each of its row sum equals one, i.e., $Qe = e$ where $e$ is the all-one vector. This paper presents a method for…

The orthogonal group acts on the space of several $n\times n$ matrices by simultaneous conjugation. For an infinite field of characteristic different from two, relations between generators for the algebra of invariants are described. As an…

We propose a superfast method for constructing orthogonal matrices $M\in\mathcal{O}(n,q)$ in finite fields $GF(q)$. It can be used to construct $n\times n$ orthogonal matrices in $Z_p$ with very high values of $n$ and $p$, and also…

We consider three families of groups: the Bianchi groups SL(2,O) where O is the ring of integers of an imaginary, quadratic field; the groups SL*(2,O) where O is a *-order of a definite, rational quaternion algebra with an orthogonal…

We study the structure of the algebra of polynomial invariants for the usual conjugation action of the complex special, SO_n, and general, O_n, orthogonal group on the space of traceless n by n complex matrices. (Note that these two…

The octonions are one of the four normed division algebras, together with the real, complex and quaternion number systems. The latter three hold a primary place in random matrix theory, where in applications to quantum physics they are…

We show that one can use model categories to construct rational orthogonal calculus. That is, given a continuous functor from vector spaces to based spaces one can construct a tower of approximations to this functor depending only on the…

Let $R$ be a finite commutative ring with unity $1_R$ and $k \in R$. Properties of one-sided $k$-orthogonal $n \times n$ matrices over $R$ are presented. When $k$ is idempotent, these matrices form a semigroup structure. Consequently new…

We present in this paper a routine which construct the ideal generated by a list of elements in a matrix Lie algebra at any particular characteristic. We have used this algorithm to analyze the problem of the simplicity of some Lie…

We show that all spin groups of non-definite, quinary quadratic forms over a field with characteristic 0 can be represented as 2 by 2 matrices with entries in an associated quaternion algebra. Over local and global fields, we further study…

A collection of subroutines and examples of their uses, as well as the underlying numerical methods, are described for generating orthogonal polynomials relative to arbitrary weight functions. The object of these routines is to produce the…

We classify finite groups $G$, such that the group algebra, $\mathbb{Q}G$ (over the field of rational numbers $\mathbb{Q}$), is the direct product of the group algebra $\mathbb{Q}[G/N]$ of a proper factor group $G/N$, and some division…

We compute the genus of a rational quadratic form in terms of the K-theory of a C*-algebra attached to the adelic orthogonal group of the form. As a corollary, one gets a higher composition law for the rational quadratic forms. As an…

Uhlenbeck proved that a set of simple elements generates the group of rational loops in GL(n,C) that satisfy the U(n)-reality condition. For an arbitrary complex reductive group, a choice of representation defines a notion of rationality…

The main question we raise here is the following one: given a real orthogonal n by n matrix X, is it true that there exists a rational orthogonal matrix Y having the same zero-pattern? We conjecture that this is the case and prove it for…

For prime powers q we use "strongly orthogonal" linear Sudoku solutions of order q^2 to construct ordered orthogonal arrays of type OOA (4,s,2,q), and for each q we present a range of values of s for which these constructions are valid.

A Formal Orthogonal Pair is a pair $(A,B)$ of symbolic rectangular matrices such that $AB^T=0$. It can be applied for the construction of Hadamard and Weighing matrices. In this paper we introduce a systematic way for constructing such…

Working over an algebraically closed base field $k$ of characteristic 2, the ring of invariants $R^G$ is studied, where $G$ is the orthogonal group O(n) or the special orthogonal group SO(n), acting naturally on the coordinate ring $R$ of…

We prove Horrocks' theorem for the odd elementary orthogonal group, which gives a decomposition of an orthogonal matrix with entries from a polynomial ring $R[X]$, over a commutative ring $R$ in which 2 is invertible, as a product of an…