$${2\times 2}$$ 2 × 2 Matrix Representation Forms and Inner Relationships of Split Quaternions

2019 ◽  
Vol 29 (2) ◽  
Author(s):  
Qiu-Ying Ni ◽  
Jin-Kou Ding ◽  
Xue-Han Cheng ◽  
Ying-Nan Jiao
Keyword(s):  
Matrix Representation ◽  
Split Quaternions

Cramer’s rule over quaternions and split quaternions: A unified algebraic approach in quaternionic and split quaternionic mechanics

2020 ◽  
pp. 2150080
Author(s):  
Gang Wang ◽  
Dong Zhang ◽  
Zhenwei Guo ◽  
Tongsong Jiang
Keyword(s):  
Matrix Representation ◽  
Linear Equations ◽  
Algebraic Approach ◽  
Complex Matrix ◽  
Split Quaternions ◽  
Quaternion Matrices ◽  
Cramer’S Rule ◽  
Algebraic Technique

This paper aims to present, in a unified manner, Cramer’s rule which are valid on both the algebras of quaternions and split quaternions. This paper, introduces a concept of v-quaternion, studies Cramer’s rule for the system of v-quaternionic linear equations by means of a complex matrix representation of v-quaternion matrices, and gives an algebraic technique for solving the system of v-quaternionic linear equations. This paper also gives a unification of algebraic techniques for Cramer’s rule in quaternionic and split quaternionic mechanics.

scholarly journals Matrices over hyperbolic split quaternions

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 913-920 ◽  
Author(s):  
Melek Erdoğdu ◽  
Mustafa Özdemir
Keyword(s):  
Matrix Representation ◽  
Quaternion Matrix ◽  
Split Quaternion ◽  
Split Quaternions ◽  
Quaternion Matrices ◽  
Hyperbolic Split Quaternion ◽  
Split Quaternion Matrix

In this paper, we present some important properties of matrices over hyperbolic split quaternions. We examine hyperbolic split quaternion matrices by their split quaternion matrix representation.

Completely simple semigroups: free products, free semigroups and varieties

1981 ◽  
Vol 88 (3-4) ◽  
pp. 293-313 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Matrix Representation ◽  
Maximal Subgroups ◽  
Lattice Of Varieties ◽  
Free Products ◽  
Completely Simple Semigroups ◽  
Countable Rank ◽  
Free Semigroups ◽  
Abelian Subgroups ◽  
And Inversion ◽  
Completely Simple

SynopsisThe class CS of completely simple semigroups forms a variety under the operations of multiplication and inversion (x−1 being the inverse of x in its ℋ-class). We determine a Rees matrix representation of the CS-free product of an arbitrary family of completely simple semigroups and deduce a description of the free completely simple semigroups, whose existence was proved by McAlister in 1968 and whose structure was first given by Clifford in 1979. From this a description of the lattice of varieties of completely simple semigroups is given in terms of certain subgroups of a free group of countable rank. Whilst not providing a “list” of identities on completely simple semigroups it does enable us to deduce, for instance, the description of all varieties of completely simple semigroups with abelian subgroups given by Rasin in 1979. It also enables us to describe the maximal subgroups of the “free” idempotent-generated completely simple semigroups T(α, β) denned by Eberhart et al. in 1973 and to show in general the maximal subgroups of the “V-free” semigroups of this type (which we define) need not be free in any variety of groups.

scholarly journals Wideband generalized admittance matrix representation for the analysis and design of waveguide filters with coaxial excitation

Radio Science ◽  
2013 ◽  
Vol 48 (1) ◽  
pp. 50-60 ◽  
Author(s):  
Fermín Mira ◽  
Ángel A. San Blas ◽  
Vicente E. Boria ◽  
Luis J. Roglá ◽  
Benito Gimeno
Keyword(s):  
Matrix Representation ◽  
Analysis And Design ◽  
Admittance Matrix ◽  
Waveguide Filters
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