On One Linear Equation in One Quaternionic Unknown

We study the quaternionic linear system which is composed out of terms of the form ln(x) := n p=1 apxbp with quaternionic constants ap, bp and a variable number n of terms. In the first place we investigate one equation in one variable. If n = 2 the corresponding equation, which is normally called Sylvester's equation will be treated completely by using only quaternionic algebra. For larger n a transition to the isomorphic (4 × 4) real matrix case is investigated. Sufficient conditions for non singularity will be obtained by using results from fixed point theorems. Connections to the Kronecker product are presented. The general case of a linear quaternionic system is treated, where each unknown is contained in a sum of the form mentioned above. As a tool the so-called column operator and its properties are used. An analogue of the Kronecker product for quaternionic systems involving terms of the form AXB is given.

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 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.

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

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.

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

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.

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

arXiv: Rings and Algebras, 2017

In this paper, we derive some necessary and sufficient solvability conditions for some systems of one sided coupled Sylvester-type real quaternion matrix equations in terms of ranks and generalized inverses of matrices. We also give the expressions of the general solutions to these systems when they are solvable. Moreover, we provide some numerical examples to illustrate our results. The findings of this paper extend some known results in the literature.

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.