In this paper, we explain how dual quaternion theory can extend to quaternions with generalized complex number (GCN) components. More specifically, algebraically examine new type and give several matrix representations both as a GCN.

If k = 1, then X is called the group inverse of A, and is denoted by X = Ag. The Drazin inverse is very useful in various applications (see, e.g. [1]–[4]; applications in singular differential and difference equations, Markov chains and iterative methods). In 1980, Cline and Greville [5] extended the Drazin inverse of square matrix to rectangular matrix, which can be generalized to the quaterni...

Nonlinear tracking control of unmanned underwater vehicles (UUVs) in 6 degrees of freedom (DOF) is discussed. The 4-parameter unit quaternion is used in a singularity-free representation of attitude. Several control laws based on a generalized Lyapunov function for the attitude dynamics are derived, including feedback from the vector quaternion, the Euler rotation vector and the Rodrigues param...

We investigate the problem of quaternion beamforming based on widely linear processing. First, a quaternion model of linear symmetric array with two-component electromagnetic (EM) vector sensors is presented. Based on array's quaternion model, we propose the general expression of a quaternion semiwidely linear (QSWL) beamformer. Unlike the complex widely linear beamformer, the QSWL beamformer i...

We show that, in quaternion quantum mechanics with a complex geometry, the minimal four Higgs of the unbroken electroweak theory naturally determine the quaternion invariance group which corresponds to the Glashow group. Consequently, we are able to identify the physical significance of the anomalous Higgs scalar solutions. We introduce and discuss the complex projection of the Lagrangian density.

a0 is called the real part of Q. An arithmetic S of Q(a, ]8) is a set of numbers having the following properties : Ca : S is closed with respect to algebraic addition. Cm: S is closed with respect to multiplication. R: For every number of 5, (4) has integral coefficients. U: 5 contains I0 , Ii and I2 (and hence I i l 2 by Cm). M : 5 is maximal ; that is, S is contained in no larger set having P...

We use the quaternion Kähler reduction technique to study old and new selfdual Einstein metrics of negative scalar curvature with at least a two-dimensional isometry group, and relate the quotient construction to the hyperbolic eigenfunction Ansatz. We focus in particular on the (semi-)quaternion Kähler quotients of (semi-)quaternion Kähler hyperboloids, analysing the completeness and topology,...

. In this paper, we propose the cyclic decomposition of factor group , and  is Z-valued class functions direct product  under operation addition, subgroup generalized characters .Then   is an abelian denoted by  where quaternion order  and cyclic  Also, find the  rational  valued  characters  table  of   when prime  numbers &...

The success of applying generalized complex orthogonal designs as space–time block codes recently motivated the definition of quaternion orthogonal designs as potential building blocks for space–timepolarization block codes. This paper offers techniques for constructing quaternion orthogonal designs via combinations of specially chosen complex orthogonal designs. One technique is used to build ...