Linear equations in quaternionic variables

2007

We study quaternionic linear equations of type λm(x) := m j=1 b j xc j = e with quaternionic constants b j , c j , e and arbitrary positive integer m. For m = 2 the resulting equation is called Sylvester's equation. For this case a complete solution (solution formula, determination of null space) will be given. For the general case we show that the solution can be found by a corresponding matrix equation of a particular simple form. This matrix form is connected with the centralizers of a quaternion and of its isomorphic image in R 4×4. We present a complete determination of these centralizers. However, the mentioned matrix form does not inlude a detection of the singular cases. The determination of singular cases is to some extent possible by applying Banach's fixed point theorem from which we are able to deduce several sufficient conditions for non singular cases. We end the paper with a conjecture on the form of the inverse of a linear mapping and show that interpolation ...

Advances in Applied Clifford Algebras, 2011

The general linear quaternionic equation with one unknown and systems of linear quaternionic equations with two unknown are solved. Examples of equations and their systems are considered.

Symmetry, 2022

In this paper, we establish the solvability conditions and the formula of the general solution to a Sylvester-like quaternion matrix equation. As an application, we give some necessary and sufficient conditions for a system of quaternion matrix equations to be consistent, and present an expression of the general solution of the system when it is solvable. We present an algorithm and an example to illustrate the main results of this paper. The findings of this paper generalize the known results in the literature.

Linear Algebra and its Applications, 2009

Dedicated to Professor Shmuel Friedland on the occasion of his 65th birthday AMS classification: 15A06 15A24 15A33 15A57 15A09

The aim is to solve a linear equation in quaternions namely, the equa- (j) and e are given quaternions, the quaternion

Linear and Multilinear Algebra, 2017

Lemma 2.6: In item (1), note that the condition JL B 1 = 0 is simpler than the corresponding condition in Corollary 2.6 of the reference [34] by virtue of the assumption GL B 1 = 0 in the lemma. Also, there is a 4th condition R F JL B 1 = 0 in Corollary 2.6 of the reference [34], but this condition is satisfied by virtue of the condition JL B 1 = 0 in the lemma.

2021

We establish the solvability conditions and the formula of the exact general solution of a significant kind of Sylvester-like quaternion matrix equation. As an application, we investigate the η-Herimitan solution of a quaternion matrix equation and give some numerical examples to illustrate the main results of this paper.

Iranian Journal of Science and Technology, Transactions A: Science, 2016

This paper studies some necessary and sufficient conditions for the existence of a solution and gives an expression of the general solution to the system of eight linear quaternion matrix equations. As an application, necessary and sufficient conditions are given for the system of certain matrix equations to have a symmetric solution. Note that in some of the mentioned conditions we use rank equalities. In addition, some numerical examples are given.

Abstract and Applied Analysis

We constitute some necessary and sufficient conditions for the system A1X1=C1, X1B1=C2, A2X2=C3, X2B2=C4, A3X1B3+A4X2B4=Cc, to have a solution over the quaternion skew field in this paper. A novel expression of general solution to this system is also established when it has a solution. The least norm of the solution to this system is also researched in this article. Some former consequences can be regarded as particular cases of this article. Finally, we give determinantal representations (analogs of Cramer’s rule) of the least norm solution to the system using row-column noncommutative determinants. An algorithm and numerical examples are given to elaborate our results.

Applied Mathematics and Computation, 2008

We in this paper establish necessary and sufficient conditions for the existence of and an explicit expression for a common solution to six classical linear quaternion matrix equations A