Representation theory is a branch of algebra that allows the study of groups through linear applications, i.e. matrices. Thus problems in abstract groups can be reduced to problems on matrices. Representation theory is the basis of character theory. In this paper we present a formalization of finite groups representation theory in the Coq system that includes a formalization of Maschke’s theore...

This paper describes new issues of the Mathematica standard package Quaternions for implementing Hamilton’s Quaternion Algebra. This work attempts to endow the original package with the ability to perform operations on symbolic expressions involving quaternion-valued functions. A collection of new functions is introduced in order to provide basic mathematical tools necessary for dealing with re...

The analysis of time series in 4D commutative hypercomplex algebras is introduced. Firstly, generalized Segre’s quaternion (GSQ) random variables and signals are studied. Then, two concepts properness suggested statistical tests to check if a GSQ vector proper or not proposed. Further, method determine which specific algebra most likely achieve, possible, the properties given. Next, both linear...

A new spacecraft attitude estimation approach based on the Unscented Filter is derived. For nonlinear systems the Unscented Filter uses a carefully selected set of sample points to more accurately map the probability distribution than the linearization of the standard Extended Kalman Filter, leading to faster convergence from inaccurate initial conditions in attitude estimation problems. The fi...

Abstract. Much of the work of Golub and his collaborators uses techniques of linear algebra to deal with problems in analysis, or employs tools from analysis to solve problems arising in linear algebra. Instances are described of such interdisciplinary work, taken from quadrature theory, orthogonal polynomials, and least squares problems on the one hand, and error analysis for linear algebraic ...

We focus on how applications that lead to large dense linear systems naturally build matrices. This allows us explain why traditional interfaces to dense linear algebra libraries for distributed memory architectures, which evolved from sequential linear algebra libraries, inherently do not support applications well. We review the application interface that has been supported by the Parallel Lin...

An extension of the FastICA algorithm is proposed for the blind separation of both Q-proper and Q-improper quaternionvalued signals. This is achieved by maximising a negentropy-based cost function, and is implemented using the Newton method in the augmented quaternion statistics framework. It is shown that the use of augmented statistics and the associated widely linear modeling provide theoret...

We present a Jacobi-like algorithm for simultaneous diagonalization of commuting pairs of complex normal matrices by unitary similarity transformations. The algorithm uses a sequence of similarity transformations by elementary complex rotations to drive the oo-diagonal entries to zero. We show that its asymptotic convergence rate is quadratic and that it is numerically stable. It preserves the ...