On the Solutions of Some Linear Complex Quaternionic Equations

Some complex quaternionic equations in the type AX − XB = C are investigated. For convenience, these equations were called generalized Sylvester-quaternion equations, which include the Sylvester equation as special cases. By the real matrix representations of complex quaternions, the necessary and sufficient conditions for the solvability and the general expressions of the solutions are obtained.


Introduction
Mathematics, as with most subjects in science and engineering, has a long and varied history. In this connection one highly significant development which occurred during the nineteenth century was the quaternions, which are the elements of noncommutative algebra. Quaternions have many important applications in many applied fields, such as computer science, quantum physics, statistic, signal, and color image processing, in rigid mechanics, quantum mechanics, control theory, and field theory; see, for example, [1].
In recent years, quaternionic equations have been investigated by many authors. For example, the author of the paper [2] classified solutions of the quaternionic equation + = .
In [3], the linear equations of the forms = and = in the real Cayley-Dickson algebras (quaternions, octonions, and sedenions) are solved and the form for the roots of such equations is established. In [4], the solutions of the equations of the forms = and = for some generalizations of quaternions and octonions are investigated. In [5], the + = linear quaternionic equation with one unknown, + = , is solved. In [6], Bolat andİpek first defined the quaternion intervals set and the quaternion interval numbers, second, they presented the vector and matrix representations for quaternion interval numbers and then investigated some algebraic properties of these representations, and finally they computed the determinant, norm, inverse, trace, eigenvalues, and eigenvectors of the matrix representation established for a quaternion interval number. In [7], the quaternionic equation + = is studied. In [8], Bolat andİpek first considered the linear octonionic equation with one unknown of the form ( ) = ( ) = = , with 0 ̸ = ∈ O; second, they presented a method which allows to reduce any octonionic equation with the left and right coefficients to a real system of eight equations and finally reached the solutions of this linear octonionic equation from this real system. In [9], Flaut and Shpakivskyi investigated the left and right real matrix representations for the complex quaternions. The theory of the quaternion equations and matrix representations of quaternions is considered completely in [2][3][4][5][6][7][8][9][10][11][12][13][14].
In this paper, we aim to obtain the solutions of some linear equations with two terms and one unknown by the method of matrix representations of complex quaternions over the complex quaternion field and to investigate the solutions of some complex quaternionic linear equations.
The paper is organized as follows. In Section 2, we start with some basic concepts and results from the theory of the quaternion equations and matrix representations of quaternions which are necessary for the following. In Section 3, we obtain the solutions of some linear equations with two terms 2 The Scientific World Journal and one unknown by the method of matrix representations of complex quaternions over the complex quaternion field. We finish the paper with some conclusions about the study presented.

Preliminaries
The following notations, definitions, propositions, lemmas, and theorems will be used to develop the proposed work. We now start the definitions of the quaternion and complex quaternion and their basic properties that will be used in the sequel.
It is well known that a complex number is a number consisting of a real and imaginary part. It can be written in the form + , where is the imaginary unit with the defining property 2 = −1. The set of all complex numbers is usually denoted by C. From here, it can be easily said that the set of complex numbers is an extension of the set of real numbers, usually denoted by R. That is, R ⊂ C.
In the literature, firstly, the set of quaternions introduced as by Irish mathematician Sir William Rowan Hamilton in 1843, is a generalized set of complex numbers. H R is an algebra over the field R, and this algebra is called the real quaternion algebra and the set {1, , , } is a basis in H R . The elements in H R take the form = 0 + 1 + 2 + 3 , where 0 , 1 , 2 , 3 ∈ R, which can simply be written as = Re + Im , where Re = 0 and Im = 1 + 2 + 3 . The conjugate of is defined as = Re −Im = 0 − 1 − 2 − 3 , which satisfies = , + = + , = for all , ∈ H R . The norm of is defined to be | | = √ = √ 2 0 + 2 1 + 2 2 + 2 3 . Some simple operation properties on quaternions are listed below: is a nonzero quaternion, Theorem 1 (see [10,15]). Let , ∈ H R be quaternions. Then and are similar if and only if 0 = 0 and 2 A complex quaternion is an element of the form = , ∈ {1, 2, 3} ; (4) and being uniquely determined by and . We denote by H C the set of the complex quaternions and H C is an algebra over the field C and this algebra is called the complex quaternion algebra. The set {1, 1 , 2 , 3 } is a basis in H C .
Throughout this note, the algebra H R is denoted by H. For the quaternion , ∈ H, if * is defined as then it satisfies the following properties: For the quaternion algebra, H, the map where = 0 0 + 1 1 + 2 2 + 3 3 ∈ H, is an isomorphism between H and the algebra of the matrices: We remark that the matrix ( ) ∈ M 4 (R) has as columns the coefficients in R of the basis {1, 1 , 2 , 3 } for the elements { , 1 , 2 , 3 }. The matrix ( ) is called the left matrix representation of the element ∈ H.
Analogously with the left matrix representation, we have for the element ∈ H the right matrix representation: The Scientific World Journal 3 We remark that the matrix ( ) ∈ M 4 (R) has as columns the coefficients in R of the basis {1, 1 , 2 , 3 } for the elements { , 1 , 2 , 3 }.

Main Results
In this section, the complex quaternionic equations in the type where Ψ( ) = ( 1 0 0 ) Θ( ) ( 1 0 0 ), which is a simple system of linear equations over R. In order to symbolically solve it, we need to examine some operation properties on the matrix Γ( ) − Ψ( ).
The results in Lemma 9 (ii) come from the following two equalities: Finally, under the conditions that 0 = 0 and | Im | = | Im |, it is easily seen that From it and a simple fact that (Im ) 2 = (Im ) 2 = −| Im | 2 , we can easily deduce the following equality: 3 ( , ) = −4| Im | 2 ( , ). So, the proof of th case in Lemma 9 is completed.
Based on Lemma 9, we have the following several results.
Theorem 10. Let = + ∈ H C be given and ∉ R. Then the general solution of the equation where ∈ H C is arbitrary.
Proof. According to (ii) and (v) cases in Proposition 8, (29) is equivalent to and since | ( , )| = 0, (31) has a nonzero solution. In that case, the general solution of (31) can be expressed as where ⃗ is an arbitrary vector in H C . Now substituting − ( , ) in Lemma 9 (iii) in it, we get (ii) in that case, the general solution of (34) is where ∈ H C is arbitrary.
Proof. According to (ii) and (v) cases in Proposition 8, (34) is equivalent to and this equation has a nonzero solution if and only if | ( , )| = 0, which is equivalent, by Lemma 9 (i), to (35).
In that case, the general solution of this equation can be expressed as where ⃗ is an arbitrary vector in H C . Now substituting − ( , ) in Lemma 9 (iii) in it, we get Returning it to complex quaternion form by (ii), (v), and (vii) in Proposition 8, we have (36).
Theorem 13. Let = + ∈ H C be given. Then the general solution of the equation where ∈ H C is arbitrary.
Proof. According to (ii) and (v) cases in Proposition 8, (42) is equivalent to and since | ( , )| = 0, (44) has a nonzero solution. In that case, the general solution of (44) can be expressed as where ⃗ is an arbitrary vector in H C . Now substituting − ( , ) in Lemma 9 (iii) in it, we get Returning it to complex quaternion form by (ii), (v), and (vii), we have (43).
In that case the general solution of (49) can be expressed as where ⃗ is an arbitrary vector in H C . Returning it to complex quaternion form, we find (48).

Conclusions
Starting from known results and referring to the real matrix representations of the complex quaternions, in this paper we have investigated solutions of some linear equations with two terms and one unknown by the method of matrix representations of complex quaternions over the complex quaternion field. The methods and results developed in this paper can also extend to complex octonionic equations. We will present them in another paper.