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.

scholarly journals Determinantal Representations of the Core Inverse and Its Generalizations with Applications

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Ivan I. Kyrchei
Keyword(s):  
Direct Method ◽  
Linear Equations ◽  
Close Relation ◽  
Numerical Example ◽  
Determinantal Representation ◽  
The Core ◽  
Core Inverse ◽  
Cramer’S Rule ◽  
The Right

In this paper, we give the direct method to find of the core inverse and its generalizations that is based on their determinantal representations. New determinantal representations of the right and left core inverses, the right and left core-EP inverses, and the DMP, MPD, and CMP inverses are derived by using determinantal representations of the Moore-Penrose and Drazin inverses previously obtained by the author. Since the Bott-Duffin inverse has close relation with the core inverse, we give its determinantal representation and its application in finding solutions of the constrained linear equations that is an analog of Cramer’s rule. A numerical example to illustrate the main result is given.

scholarly journals A Representation of Nonhomogeneous Quadratic Forms with Application to the Least Squares Solution

2011 ◽  
Vol 2011 ◽  
pp. 1-5 ◽  
Author(s):  
Czesław Stępniak
Keyword(s):  
Least Squares ◽  
Quadratic Forms ◽  
Differential Calculus ◽  
Linear Models ◽  
Linear Equations ◽  
Algebraic Approach ◽  
System Of Linear Equations ◽  
Least Squares Solution ◽  
Inconsistent System ◽  
Usual Solution

The least squares problem appears, among others, in linear models, and it refers to inconsistent system of linear equations. A crucial question is how to reduce the least squares solution in such a system to the usual solution in a consistent one. Traditionally, this is reached by differential calculus. We present a purely algebraic approach to this problem based on some identities for nonhomogeneous quadratic forms.

scholarly journals A novel interpretation of least squares solution

1992 ◽  
Vol 15 (1) ◽  
pp. 41-46
Author(s):  
Jack-Kang Chan
Keyword(s):  
Least Squares ◽  
Source Localization ◽  
Convex Combination ◽  
Linear Equations ◽  
Singular Values ◽  
Statistical Interpretation ◽  
System Of Linear Equations ◽  
Overdetermined System ◽  
The Matrix ◽  
Cramer’S Rule

We show that the well-known least squares (LS) solution of an overdetermined system of linear equations is a convex combination of all the non-trivial solutions weighed by the squares of the corresponding denominator determinants of the Cramer's rule. This Least Squares Decomposition (LSD) gives an alternate statistical interpretation of least squares, as well as another geometric meaning. Furthermore, when the singular values of the matrix of the overdetermined system are not small, the LSD may be able to provide flexible solutions. As an illustration, we apply the LSD to interpret the LS-solution in the problem of source localization.

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