Research Article | Open Access

# On the Solutions of Some Linear Complex Quaternionic Equations

**Academic Editor:**José Carlos Costa

#### Abstract

Some complex quaternionic equations in the type 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.

#### 1. 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 and İpek 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 and İpek first considered the linear octonionic equation with one unknown of the form , with ; 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 [2–14].

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 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.

#### 2. 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 . The set of all complex numbers is usually denoted by . 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 . That is, .

In the literature, firstly, the set of quaternions introduced as with by Irish mathematician Sir William Rowan Hamilton in 1843, is a generalized set of complex numbers. is an algebra over the field , and this algebra is called the real quaternion algebra and the set is a basis in . The elements in take the form , where , which can simply be written as , where and . The conjugate of is defined as , which satisfies for all . The norm of is defined to be . Some simple operation properties on quaternions are listed below:

Theorem 1 (see [10, 15]). *Let be quaternions. Then and are similar if and only if and , that is, and .*

A complex quaternion is an element of the form , where , and being uniquely determined by and . We denote by the set of the complex quaternions and is an algebra over the field and this algebra is called the complex quaternion algebra. The set is a basis in .

The element , , can be written as where , , and . Therefore, we can write a complex quaternion as the form , where and are in . The conjugate of the complex quaternion is the element , and it satisfies Throughout this note, the algebra is denoted by . For the quaternion , if is defined as then it satisfies the following properties: For the quaternion algebra, , the map where , is an isomorphism between and the algebra of the matrices: We remark that the matrix has as columns the coefficients in of the basis for the elements . The matrix is called the left matrix representation of the element .

Analogously with the left matrix representation, we have for the element the right matrix representation: where .

We remark that the matrix has as columns the coefficients in of the basis for the elements .

Proposition 2 (see [12]). *For and , one has*(i),
,
*, *,
(ii),
,
*, *,
(iii),
,
* where ** is the inverse of ** nonzero quaternion.*

Proposition 3 (see [12]). *For , let be the vector representation of the element . Therefore for all the following relations are fulfilled:*(i)*,*(ii)*,*(iii)*,*(iv)*,*(v)*, where * * and it is the weak norm of *.

For details about the matrix representations of the real quaternions, the reader is referred to [12].

The matrix
where is a complex quaternion, with , **,** and , is called the left real matrix representation for the complex quaternion . The right real matrix representation for the complex quaternion is the matrix:
We remark that , ; see [9].

Proposition 4 (see [9]). *Let be two quaternions; then, the following relations are true.*(i)*, where *. (ii)*, where *. (iii)*, where *. (iv)*.*(v)*For *,

Proposition 5 (see [9]). *Let be given. Then
*

Proposition 6 (see [9]). *Let be given. Then
*

*Definition 7 (see [9]). *Let , be given. Then
is the vector representation of the element , where and , are the vector representations of the quaternions and .

Proposition 8 (see [9]). *Let , be given. Then*(i),
* where ** is the identity matrix and ** is the zero matrix;*(ii)*;*(iii),
* where **;*(iv)*;*(v)*;*(vi)*;*(vii)*, where **.*

#### 3. Main Results

In this section, the complex quaternionic equations in the type are considered. Using the representation matrices and of complex quaternions, the necessary and sufficient conditions for the solvability and the general expression of the solutions are obtained.

According to (ii) and (v) cases in Proposition 8, (18) is equivalent to where , which is a simple system of linear equations over . In order to symbolically solve it, we need to examine some operation properties on the matrix .

Lemma 9. *Let be given, and denote . Then*(i)*the determinant of is
where .*(ii)*if , or , then is nonsingular and its inverse can be expressed as
or
*(iii)*if and , then is singular and has a generalized inverse as follows:
*

*Proof. *It is a known result that, for all , , there are nonzero such that
where and . Now applying Propositions 5, 6, and 8 to both of them we obtain
Thus we can derive
Consequently, substituting and into it, the proof of th case in Lemma 9 is completed.

The results in Lemma 9 (ii) come from the following two equalities:

Finally, under the conditions that and , it is easily seen that
From it and a simple fact that , we can easily deduce the following equality: . So, the proof of th case in Lemma 9 is completed.

Based on Lemma 9, we have the following several results.

Theorem 10. *Let be given and . Then the general solution of the equation
**
is
**
where is arbitrary.*

*Proof. *According to (ii) and (v) cases in Proposition 8, (29) is equivalent to
and since , (31) has a nonzero solution. In that case, the general solution of (31) can be expressed as
where is an arbitrary vector in . 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 (30).

Theorem 11. *Let be given. Then*(i)*the linear equation
has a nonzero solution; that is, and are similar, if and only if
*(ii)*in that case, the general solution of (34) is
where 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 , 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 . 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 12. *Let , , be given with and being not similar; that is, or . Then (18) has a unique solution
*

*Proof. *Under the assumption of this theorem, is nonsingular by Lemma 9 (ii). Hence (19) has a unique solution as follows:
and the result follows using (ii) and (v) of Proposition 8.

Theorem 13. *Let be given. Then the general solution of the equation
**
is
**
where is arbitrary.*

*Proof. *According to (ii) and (v) cases in Proposition 8, (42) is equivalent to
and since , (44) has a nonzero solution. In that case, the general solution of (44) can be expressed as
where is an arbitrary vector in . Now substituting in Lemma 9 (iii) in it, we get
Returning it to complex quaternion form by (ii), (v), and (vii), we have (43).

Theorem 14. *Let , , be given with . Then (18) has a solution if and only if
*

*in which case the general solution of (18) can be written as*

*where is arbitrary.*

*Proof. *According to (ii) and (v) cases in Proposition 8, (18) is equivalent to
This equation is solvable if and only if
which is equivalent to
Returning it to complex quaternion form by (ii) and (v) cases in Proposition 8, we obtain
which is equivalent to , and then (47). In that case the general solution of (49) can be expressed as
where is an arbitrary vector in . Returning it to complex quaternion form, we find (48).

#### 4. 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.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This study is a part of the corresponding author's Ph.D. thesis. The authors would like to thank the anonymous referees for their careful reading and valuable suggestions which improved this work.

#### References

- S. L. Adler,
*Quaternionic Quantum Mechanics and Quantum Fields*, Oxford University Press, New York, NY, USA, 1994. - R. M. Porter, “Quaternionic linear and quadratic equations,”
*Journal of Natural Geometry*, vol. 11, no. 2, pp. 101–106, 1997. View at: Google Scholar | MathSciNet - Y. Tian, “Similarity and consimilarity of elements in the real Cayley-Dickson algebras,”
*Advances in Applied Clifford Algebras*, vol. 9, no. 1, pp. 61–76, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Flaut, “Some equation in algebras obtained by Cayley-Dickson process,”
*Analele Stiintifice ale Universitatii Ovidius Constanta*, vol. 9, no. 2, pp. 45–68, 2001. View at: Google Scholar - V. S. Shpakivskyi, “Linear quaternionic equations and their systems,”
*Advances in Applied Clifford Algebras*, vol. 21, no. 3, pp. 637–645, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Bolat and A. Ipek, “Quaternion interval numbers and a matrix approach to them,”
*Transylvanian Journal of Mathematics and Mechanics*, vol. 2, no. 2, pp. 131–139, 2010. View at: Google Scholar | MathSciNet - J. Helmstetter, “The quaternionic equation $ax+xb=c$,”
*Advances in Applied Clifford Algebras*, vol. 22, no. 4, pp. 1055–1059, 2012. View at: Publisher Site | Google Scholar | MathSciNet - C. Bolat and A. Ipek, “A method to find the solution of the linear octonionic equation $\alpha \left(x\alpha \right)=\left(\alpha x\right)\alpha =\alpha x\alpha =\rho $,”
*General Mathematics Notes*, vol. 12, no. 2, pp. 10–18, 2012. View at: Google Scholar - C. Flaut and V. Shpakivskyi, “Real matrix representations for the complex quaternions,”
*Advances in Applied Clifford Algebras*, vol. 23, no. 3, pp. 657–671, 2013. View at: Publisher Site | Google Scholar | MathSciNet - J. L. Brenner, “Matrices of quaternions,”
*Pacific Journal of Mathematics*, vol. 1, pp. 329–335, 1951. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. E. Johnson, “On the equation $\chi \alpha =\gamma \chi +\beta $ over an algebraic division ring,”
*Bulletin of the American Mathematical Society*, vol. 50, pp. 202–207, 1944. View at: Publisher Site | Google Scholar | MathSciNet - Y. Tian, “Matrix representations of octonions and their applications,”
*Advances in Applied Clifford Algebras*, vol. 10, no. 1, pp. 61–90, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Tian, “Universal factorization equalities for quaternion matrices and their applications,”
*Mathematical Journal of Okayama University*, vol. 41, pp. 45–62 (2001), 1999. View at: Google Scholar | Zentralblatt MATH | MathSciNet - F. Zhang, “Quaternions and matrices of quaternions,”
*Linear Algebra and Its Applications*, vol. 251, pp. 21–57, 1997. View at: Publisher Site | Google Scholar | MathSciNet - Y. H. Au-Yeung, “On the convexity of numerical range in quaternionic Hilbert spaces,”
*Linear and Multilinear Algebra*, vol. 16, no. 1–4, pp. 93–100, 1984. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2014 Cennet Bolat and Ahmet İpek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.