On the Solutions of Some Linear Complex Quaternionic Equations

Abstract

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.

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 𝑎𝑥+𝑥𝑏=𝑐. In [3], the linear equations of the forms 𝑎𝑥=𝑥𝑏 and 𝑎𝑥=𝑥𝑏 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 𝑎𝑥=𝑥𝑏 and 𝑎𝑥=𝑥𝑏 for some generalizations of quaternions and octonions are investigated. In [5], the 𝛼𝑥𝛽+𝛾𝑥𝛿=𝜌 linear quaternionic equation with one unknown, 𝛼𝑥𝛽+𝛾𝑥𝛿=𝜌, 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 𝑎𝑥+𝑥𝑏=𝑐 is studied. In [8], Bolat and İpek first considered the linear octonionic equation with one unknown of the form 𝛼(𝑥𝛼)=(𝛼𝑥)𝛼=𝛼𝑥𝛼=𝜌, with 0≠𝛼∈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 [2–14].

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 𝑎+𝑏𝑖, where 𝑖 is the imaginary unit with the defining property 𝑖²=−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 𝐇_ℝ=𝑎=𝑎₀+𝑎₁𝑖+𝑎₂𝑗+𝑎₃𝑘∶𝑎_𝑠∈ℝ,𝑠=0,1,2,3, (1) with 𝑖²=𝑗²=𝑘²=−1,𝑖𝑗=𝑘=−𝑗𝑖,𝑗𝑘=𝑖=−𝑘𝑗,𝑘𝑖=𝑗=−𝑖𝑘,𝑖𝑗𝑘=−1, (2) 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,𝑖,𝑗,𝑘} is a basis in 𝐇_ℝ. The elements in 𝐇_ℝ take the form 𝑎=𝑎₀+𝑎₁𝑖+𝑎₂𝑗+𝑎₃𝑘, where 𝑎₀,𝑎₁,𝑎₂,𝑎₃∈ℝ, which can simply be written as 𝑎=Re𝑎+Im𝑎, where Re𝑎=𝑎₀ and Im𝑎=𝑎₁𝑖+𝑎₂𝑗+𝑎₃𝑘. The conjugate of 𝑎 is defined as 𝑎=Re𝑎−Im𝑎=𝑎₀−𝑎₁𝑖−𝑎₂𝑗−𝑎₃𝑘, which satisfies 𝑎=𝑎,𝑎+𝑏=𝑎+𝑏,𝑎𝑏=𝑏𝑎 for all 𝑎,𝑏∈𝐇_ℝ. The norm of 𝑎 is defined to be |𝑎|=√𝑎𝑎=𝑎²₀+𝑎²₁+𝑎²₂+𝑎²₃. Some simple operation properties on quaternions are listed below: 𝑎²−2(Re𝑎)𝑎+|𝑎|²()=0,Im𝑎²=−Im𝑎²,||=|𝑎𝑏|,𝑎𝑎𝑏||₋₁=𝑎|𝑎|²()().,𝑎isanonzeroquaternion,Re𝑎𝑏=Re𝑏𝑎 (3)

Theorem 1 (see [10, 15]). Let 𝑎,𝑏∈𝐇_ℝ be quaternions. Then 𝑎 and 𝑏 are similar if and only if 𝑎₀=𝑏₀ and 𝑎²₁+𝑎²₂+𝑎²₃=𝑏²₁+𝑏²₂+𝑏²₃, that is, Re(𝑎)=Re(𝑏) and |Im𝑎|²=|Im𝑏|².

A complex quaternion is an element of the form 𝑄=𝑐₀𝑒₀+𝑐₁𝑒₁+𝑐₂𝑒₂+𝑐₃𝑒₃, where 𝑐_𝑛∈ℂ,𝑛∈{0,1,2,3}, 𝑒²_𝑛{},𝑒=−1,𝑛∈1,2,3_𝑚𝑒_𝑛=−𝑒_𝑛𝑒_𝑚=𝛽_𝑚𝑛𝑒_𝑡,𝛽_𝑚𝑛∈{−1,1},𝑚≠𝑛,𝑚,𝑛∈{1,2,3}; (4)𝛽_𝑚𝑛 and 𝑒_𝑡 being uniquely determined by 𝑒_𝑚 and 𝑒_𝑛. 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,𝑒₁,𝑒₂,𝑒₃} is a basis in 𝐇_ℂ.

The element 𝑄=𝑐₀𝑒₀+𝑐₁𝑒₁+𝑐₂𝑒₂+𝑐₃𝑒₃∈𝐇_ℂ,  𝑐_𝑚∈ℂ,𝑚∈{0,1,2,3}, can be written as 𝑄=𝑎₀+𝑖𝑏₀+𝑎₁+𝑖𝑏₁𝑒₁+𝑎₂+𝑖𝑏₂𝑒₂+𝑎₃+𝑖𝑏₃𝑒₃, (5) where 𝑎_𝑚,𝑏_𝑚∈ℝ,  𝑚∈{0,1,2,3}, and 𝑖²=−1. Therefore, we can write a complex quaternion as the form 𝑄=𝑎+𝑖𝑏, where 𝑎=𝑎₀+𝑎₁𝑒₁+𝑎₂𝑒₂+𝑎₃𝑒₃ and 𝑏=𝑏₀+𝑏₁𝑒₁+𝑏₂𝑒₂+𝑏₃𝑒₃ are in 𝐇_ℝ. The conjugate of the complex quaternion 𝑄 is the element 𝑄=𝑐₀𝑒₀−𝑐₁𝑒₁−𝑐₂𝑒₂−𝑐₃𝑒₃, and it satisfies 𝑄=𝑎+𝑖𝑏. (6) Throughout this note, the algebra 𝐇_ℝ is denoted by 𝐇. For the quaternion 𝑎,𝑏∈𝐇, if 𝑎^∗ is defined as 𝑎^∗=𝑎₀+𝑎₁𝑒₁−𝑎₂𝑒₂−𝑎₃𝑒₃, (7) then it satisfies the following properties: 𝑎^∗∗=𝑎,(𝑎+𝑏)^∗=𝑎^∗+𝑏^∗. (8) For the quaternion algebra, 𝐇, the map 𝜆∶𝐇⟶𝐌₄(ℝ)(𝑎)=𝑎,𝜆⁰−𝑎₁−𝑎₂−𝑎₃𝑎₁𝑎₀−𝑎₃𝑎₂𝑎₂𝑎₃𝑎₀−𝑎₁𝑎₃−𝑎₂𝑎₁𝑎₀, (9) where 𝑎=𝑎₀𝑒₀+𝑎₁𝑒₁+𝑎₂𝑒₂+𝑎₃𝑒₃∈𝐇, is an isomorphism between 𝐇 and the algebra of the matrices: 𝐌₄(ℝ)=𝑎₀−𝑎₁−𝑎₂−𝑎₃𝑎₁𝑎₀−𝑎₃𝑎₂𝑎₂𝑎₃𝑎₀−𝑎₁𝑎₃−𝑎₂𝑎₁𝑎₀,𝑎₀,𝑎₁,𝑎₂,𝑎₃∈ℝ. (10) We remark that the matrix 𝜆(𝑎)∈𝐌₄(ℝ) has as columns the coefficients in ℝ of the basis {1,𝑒₁,𝑒₂,𝑒₃} for the elements {𝑎,𝑎𝑒₁,𝑎𝑒₂,𝑎𝑒₃}. The matrix 𝜆(𝑎) is called the left matrix representation of the element 𝑎∈𝐇.

Analogously with the left matrix representation, we have for the element 𝑎∈𝐇 the right matrix representation: 𝜌∶𝐇⟶𝐌₄(ℝ)(𝑎)=𝑎,𝜌⁰−𝑎₁−𝑎₂−𝑎₃𝑎₁𝑎₀𝑎₃−𝑎₂𝑎₂−𝑎₃𝑎₀−𝑎₁𝑎₃𝑎₂−𝑎₁𝑎₀, (11) where 𝑎=𝑎₀𝑒₀+𝑎₁𝑒₁+𝑎₂𝑒₂+𝑎₃𝑒₃∈𝐇.

We remark that the matrix 𝜌(𝑎)∈𝐌₄(ℝ) has as columns the coefficients in ℝ of the basis {1,𝑒₁,𝑒₂,𝑒₃} for the elements {𝑎,𝑒₁𝑎,𝑒₂𝑎,𝑒₃𝑎}.

Proposition 2 (see [12]). For 𝑥,𝑦∈𝐇 and 𝑟∈ℝ, one has(i)𝜆(𝑥+𝑦)=𝜆(𝑥)+𝜆(𝑦), 𝜆(𝑥𝑦)=𝜆(𝑥)𝜆(𝑦), 𝜆(𝑟𝑥)=𝑟𝜆(𝑥), 𝜆(1)=𝐼₄, (ii)𝜌(𝑥+𝑦)=𝜌(𝑥)+𝜌(𝑦), 𝜌(𝑥𝑦)=𝜌(𝑦)𝜌(𝑥), 𝜌(𝑟𝑥)=𝑟𝜌(𝑥), 𝜌(1)=𝐼₄, (iii)𝜆(𝑥⁻¹)=(𝜆(𝑥))⁻¹, 𝜌(𝑥⁻¹)=(𝜌(𝑥))⁻¹, where 𝑥⁻¹ is the inverse of 𝑥 nonzero quaternion.

Proposition 3 (see [12]). For 𝑥∈𝐇, let 𝑥=(𝑎₀,𝑎₁,𝑎₂,𝑎₃)^𝑡 ∈𝐌_1×4(ℝ) be the vector representation of the element 𝑥. Therefore for all 𝑎,𝑏,𝑥∈𝐇 the following relations are fulfilled:(i)𝑎𝑥=𝜆(𝑎)𝑥,(ii)𝑥𝑏=𝜌(𝑏)𝑥,(iii)𝑎𝑥𝑏=𝜆(𝑎)𝜌(𝑏)𝑥=𝜌(𝑏)𝜆(𝑎)𝑥,(iv)𝜌(𝑏)𝜆(𝑎)=𝜆(𝑎)𝜌(𝑏),(v)det(𝜆(𝑥))=det(𝜌(𝑥))=(𝑛(𝑥))², where 𝑛(𝑥)= 𝑎²₀+𝑎²₁+𝑎²₂+𝑎²₃ and it is the weak norm of 𝑥.

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

The matrix Γ(𝑄)=𝜆(𝑎)−𝜆𝑏^∗𝜆(𝑏)𝜆𝑎^∗, (12) where 𝑄=𝑎+𝑖𝑏 is a complex quaternion, with 𝑎=𝑎₀𝑒₀+𝑎₁𝑒₁+𝑎₂𝑒₂+𝑎₃𝑒₃∈𝐇,  𝑏=𝑏₀𝑒₀+𝑏₁𝑒₁+𝑏₂𝑒₂+𝑏₃𝑒₃∈𝐇, and 𝑖²=−1, is called the left real matrix representation for the complex quaternion 𝑄. The right real matrix representation for the complex quaternion 𝑄 is the matrix: Θ(𝑄)=𝜌(𝑎)(𝑏)𝜌𝑏−𝜌_∗𝜌𝑎^∗. (13) We remark that Γ(𝑄),  Θ(𝑄)∈𝐌₈(ℝ); see [9].

Proposition 4 (see [9]). Let 𝑎,𝑥∈𝐇 be two quaternions; then, the following relations are true.(i)𝑎^∗𝑖=𝑖𝑎, where 𝑖²=−1. (ii)𝑎𝑖=𝑖𝑎^∗, where 𝑖²=−1. (iii)−𝑎^∗=𝑖𝑎𝑖, where 𝑖²=−1. (iv)(𝑥𝑎)^∗=𝑥^∗𝑎^∗.(v)For 𝑋,𝐴∈𝐇_ℂ,𝑋=𝑥+𝑖𝑦,𝐴=𝑎+𝑖𝑏, 𝑋𝐴=𝑥𝑎−𝑦^∗𝑏+𝑖𝑥^∗𝑏+𝑦𝑎. (14)

Proposition 5 (see [9]). Let 𝑋,𝐴∈𝐇_ℂ,𝑋=𝑥+𝑖𝑦,𝐴=𝑎+𝑖𝑏 be given. Then Γ(𝑋𝐴)=Γ(𝑋)Γ(𝐴). (15)

Proposition 6 (see [9]). Let 𝑋,𝐴∈𝐇_ℂ,𝑋=𝑥+𝑖𝑦,𝐴=𝑎+𝑖𝑏 be given. Then Θ(𝑋𝐴)=Θ(𝐴)Θ(𝑋). (16)

Definition 7 (see [9]). Let 𝑋∈𝐇_ℂ,  𝑋=𝑥+𝑖𝑦 be given. Then 𝑋=𝑥,𝑦^𝑡∈𝐌_8×1(ℝ) (17) is the vector representation of the element 𝑋, where 𝑥,𝑦∈𝐇 and 𝑥=(𝑥₀,𝑥₁,𝑥₂,𝑥₃)^𝑡∈𝐌_4×1(ℝ),𝑦=(𝑦₀,𝑦₁,𝑦₂,𝑦₃)^𝑡∈𝐌_4×1(ℝ) are the vector representations of the quaternions 𝑥 and 𝑦.

Proposition 8 (see [9]). Let 𝑋,𝐴,𝐵∈𝐇_ℂ,𝑋=𝑥+𝑖𝑦,  𝑥,𝑦∈𝐇 be given. Then(i)𝑋=Γ(𝑋)¹₀, where 1=𝐼₄∈𝐌₄(ℝ) is the identity matrix and 0=𝑂₄∈𝐌₄(ℝ) is the zero matrix;(ii)𝐴𝑋=Γ(𝐴)𝑋;(iii)𝛼𝑦^∗=𝑦, where 𝛼=^10000100_{00−10000−1}∈𝐌₄(ℝ);(iv)𝛼²=𝐼₄;(v)𝑋𝐴=¹⁰_0𝛼Θ(𝐴)¹⁰_0𝛼𝑋;(vi)Γ(𝐴)¹⁰_0𝛼Θ(𝐵)¹⁰_0𝛼=¹⁰_0𝛼Θ(𝐵)¹⁰_0𝛼Γ(𝐴);(vii)𝐴𝑋𝐵=Γ(𝐴)Ψ(𝐵)𝑋, where Ψ(𝐵)=¹⁰_0𝛼Θ(𝐵)¹⁰_0𝛼.

3. Main Results

In this section, the complex quaternionic equations in the type 𝐴𝑋−𝑋𝐵=𝐶 (18) are considered. Using the representation matrices Γ(⋅) and Θ(⋅) 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 [Γ(𝐴)−Ψ(𝐵)]𝑋=𝐶, (19) where Ψ(𝐵)=¹⁰_0𝛼Θ(𝐵)¹⁰_0𝛼, 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(𝐴)−Ψ(𝐵).

Lemma 9. Let 𝐴=𝑎+𝑖𝑏,𝐵=𝑐+𝑖𝑑∈𝐇_ℂ be given, and denote 𝛿(𝐴,𝐵)=Γ(𝐴)−Ψ(𝐵). Then(i)the determinant of 𝛿(𝐴,𝐵) is |𝛿(=𝑠𝐴,𝐵)|₂+(|Im𝐴|−|Im𝐵|)²𝑠²+(|Im𝐴|+|Im𝐵|)²²=𝑠⁴+2𝑠²|Im𝐴|²+|Im𝐵|²+|Im𝐴|²−|Im𝐵|²²², (20) where 𝑠=𝐴₀−𝐵₀.(ii)if 𝐴₀≠𝐵₀, or |Im𝐴|≠|Im𝐵|, then 𝛿(𝐴,𝐵) is nonsingular and its inverse can be expressed as 𝛿⁻¹(𝐴,𝐵)=Γ⁻¹|𝐵|²−|𝐴|²+2𝐴₀−𝐵₀𝐴×Γ(𝐴)−Ψ𝐵; (21) or 𝛿⁻¹(𝐴,𝐵)=Ψ⁻¹|𝐴|²−|𝐵|²+2𝐵₀−𝐴₀𝐵×Γ𝐴−Ψ(𝐵); (22)(iii)if 𝐴₀=𝐵₀ and |Im𝐴|=|Im𝐵|, then 𝛿(𝐴,𝐵) is singular and has a generalized inverse as follows: 𝛿⁻()1𝐴,𝐵=−4|Im𝐴|²𝛿()1𝐴,𝐵=−4|Im𝐴|²[Γ(Im𝐴)−Ψ(Im𝐵)]. (23)

Proof. It is a known result that, for all 𝐴,  𝐵∈𝐇_ℂ, there are nonzero 𝑃,𝑄∈𝐇_ℂ such that 𝐴=𝑃𝐴₀+|Im𝐴|𝑖𝑃⁻¹=𝑃𝐴𝑃⁻¹,𝐵=𝑄𝐵₀+|Im𝐵|𝑖𝑄⁻¹=𝑄𝐵𝑄⁻¹, (24) where 𝐴=𝐴₀+|Im𝐴|𝑖 and 𝐵=𝐵₀+|Im𝐵|𝑖. Now applying Propositions 5, 6, and 8 to both of them we obtain Γ(𝐴)=Γ(𝑃)Γ𝐴Γ𝑃⁻¹,Ψ(𝐵)=Ψ𝑄⁻¹Ψ𝐵Ψ(𝑄). (25) Thus we can derive |𝛿(=|Γ(𝐴)(𝐵=Γ(𝑃)Γ𝐴Γ𝑃𝐴,𝐵)|−Ψ)|₋₁−Ψ𝑄⁻¹Ψ𝐵Ψ(𝑄=|Γ(𝑃Γ𝐴𝑃))|−Γ⁻¹Ψ×𝑄⁻¹Ψ𝐵Ψ(𝑄)Γ(𝑃×Γ𝑃₋₁)=Γ𝐴−Ψ𝑄⁻¹Ψ𝐵Ψ(𝑄=Ψ𝑄₋₁)Ψ(𝑄)Γ𝐴Ψ𝑄⁻¹𝐵Ψ(𝑄=Γ𝐴𝐵.−Ψ|)|−Ψ (26) Consequently, substituting 𝐴=𝐴₀+|Im𝐴|𝑖 and 𝐵=𝐵₀+|Im𝐵|𝑖 into it, the proof of 𝑖th case in Lemma 9 is completed.
The results in Lemma 9 (ii) come from the following two equalities: Γ(𝐴)−Ψ𝐵Γ(𝐴)(𝐵𝐴[−Ψ)]=Γ²+|𝐵|²𝐼₈−2𝐵₀Γ(𝐴)=Γ𝐴²−2𝐵₀𝐴+|𝐵|²,Γ𝐴(𝐵Γ(𝐴)(𝐵𝐵−Ψ)[−Ψ)]=Ψ²+|𝐴|²𝐼₈−2𝐴₀Ψ(𝐵)=Ψ𝐵²+|𝐴|²−2𝐴₀𝐵. (27)
Finally, under the conditions that 𝐴₀=𝐵₀ and |Im𝐴|=|Im𝐵|, it is easily seen that 𝛿(𝐴,𝐵)=Γ(𝐴)−Ψ(𝐵)=Γ(Im𝐴)−Ψ(Im𝐵). (28) From it and a simple fact that (Im𝐴)²=(Im𝐵)²=−|Im𝐴|², we can easily deduce the following equality: 𝛿³(𝐴,𝐵)=−4|Im𝐴|²𝛿(𝐴,𝐵). So, the proof of 𝑖𝑖𝑖th case in Lemma 9 is completed.

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

Theorem 10. Let 𝐴=𝑎+𝑖𝑏∈𝐇_ℂ be given and 𝐴∉ℝ. Then the general solution of the equation 𝐴𝑋=𝑋𝐴 (29) is 1𝑋=𝑃−|Im𝐴|²(Im𝐴)𝑃(Im𝐴), (30) where 𝑃∈𝐇_ℂ is arbitrary.

Proof. According to (ii) and (v) cases in Proposition 8, (29) is equivalent to [Γ(𝐴)−Ψ(𝐴)]𝑋=𝛿(𝐴,𝐴)𝑋=0, (31) and since |𝛿(𝐴,𝐴)|=0, (31) has a nonzero solution. In that case, the general solution of (31) can be expressed as 𝑋=2𝐼₈−𝛿⁻(𝐴,𝐴)𝛿(𝐴,𝐴)𝑃, (32) where 𝑃 is an arbitrary vector in 𝐇_ℂ. Now substituting 𝛿⁻(𝐴,𝐴) in Lemma 9 (iii) in it, we get 𝑋=2𝐼₈+14|Im𝐴|²𝛿²(𝑃𝐼𝐴,𝐴)=2₈+14|Im𝐴|²×−2|Im𝐴|²𝐼₈()Ψ(𝑃=𝐼+ΓIm𝐴Im𝐴)₈−1|Im𝐴|²Γ(Im𝐴)Ψ(Im𝐴)𝑃. (33) Returning it to complex quaternion form by (ii), (v), and (vii) in Proposition 8, we have (30).

Theorem 11. Let 𝐴=𝑎+𝑖𝑏,𝐵=𝑐+𝑖𝑑∈𝐇_ℂ be given. Then(i)the linear equation 𝐴𝑋=𝑋𝐵 (34) has a nonzero solution; that is, 𝐴 and 𝐵 are similar, if and only if Re𝐴=Re𝐵,|Im𝐴|=|Im𝐵|; (35)(ii)in that case, the general solution of (34) is 1𝑋=𝑃−|Im𝐴|²(Im𝐴)𝑃(Im𝐵), (36) where 𝑃∈𝐇_ℂ is arbitrary.

Proof. According to (ii) and (v) cases in Proposition 8, (34) is equivalent to [Γ(𝐴)−Ψ(𝐵)]𝑋=𝛿(𝐴,𝐵)𝑋=0, (37) and this equation has a nonzero solution if and only if |𝛿(𝐴,𝐵)|=0, which is equivalent, by Lemma 9 (i), to (35). In that case, the general solution of this equation can be expressed as 𝑋=2𝐼₈−𝛿⁻(𝐴,𝐵)𝛿(𝐴,𝐵)𝑃, (38) where 𝑃 is an arbitrary vector in 𝐇_ℂ. Now substituting 𝛿⁻(𝐴,𝐴) in Lemma 9 (iii) in it, we get 𝑋=2𝐼₈−𝛿⁻()𝛿(𝑃𝐼𝐴,𝐵𝐴,𝐵)=2₈+14|Im𝐴|²𝛿²(𝑃𝐼𝐴,𝐵)=2₈+1|Im𝐴|²−2|Im𝐴|²𝐼₈()Ψ(𝑃=𝐼+ΓIm𝐴Im𝐵)₈−1|Im𝐴|²Γ(Im𝐴)Ψ(Im𝐵)𝑃. (39) Returning it to complex quaternion form by (ii), (v), and (vii) in Proposition 8, we have (36).

Theorem 12. Let 𝐴=𝑎+𝑖𝑏,𝐵=𝑐+𝑖𝑑∈𝐇_ℂ,  𝑎,𝑏,𝑐,𝑑∈𝐇, be given with 𝐴 and 𝐵  being not similar; that is, Re𝐴≠Re𝐵 or |Im𝐴|≠|Im𝐵|. Then (18) has a unique solution 𝑋=2𝐴₀−𝐵₀𝐴−|𝐴|²+|𝐵|²⁻¹𝐴𝐶−𝐶𝐵=𝐴𝐶−𝐶𝐵2𝐵₀−𝐴₀𝐵+|𝐴|²−|𝐵|²⁻¹. (40)

Proof. Under the assumption of this theorem, 𝛿(𝐴,𝐵)=Γ(𝐴)−Ψ(𝐵) is nonsingular by Lemma 9 (ii). Hence (19) has a unique solution as follows: 𝑋=𝛿⁻¹(𝐴,𝐵)𝐶=Γ⁻¹2𝐴₀−𝐵₀𝐴−|𝐴|²+|𝐵|²×Γ(𝐴)−Ψ𝐵𝐶𝑋=𝛿⁻¹(𝐴,𝐵)𝐶=Γ⁻¹2𝐴₀−𝐵₀𝐴−|𝐴|²+|𝐵|²×Γ𝐴−Ψ(𝐵)𝐶, (41) and the result follows using (ii) and (v) of Proposition 8.

Theorem 13. Let 𝐴=𝑎+𝑖𝑏∈𝐇_ℂ be given. Then the general solution of the equation 𝐴𝑋−𝑋𝐴=𝐶 (42) is 1𝑋=4|Im𝐴|²()1𝐶𝐴−𝐴𝐶+𝑃−|Im𝐴|²(Im𝐴)𝑃(Im𝐴), (43) where 𝑃∈𝐇_ℂ is arbitrary.

Proof. According to (ii) and (v) cases in Proposition 8, (42) is equivalent to [Γ(𝐴)−Ψ(𝐴)]𝑋=𝛿(𝐴,𝐴)𝑋=𝐶, (44) and since |𝛿(𝐴,𝐴)|=0, (44) has a nonzero solution. In that case, the general solution of (44) can be expressed as 𝑋=𝛿⁻(𝐴,𝐴)𝐶+2𝐼₈−𝛿⁻(𝐴,𝐴)𝛿(𝐴,𝐴)𝑃, (45) where 𝑃 is an arbitrary vector in 𝐇_ℂ. Now substituting 𝛿⁻(𝐴,𝐴) in Lemma 9 (iii) in it, we get 𝑋=𝛿⁻(𝐴,𝐴)𝐶+2𝐼₈−𝛿⁻()𝛿(𝑃1𝐴,𝐴𝐴,𝐴)=−4|Im𝐴|²𝛿(𝐴,𝐴)𝐶+2𝐼₈−𝛿⁻()𝛿(𝑃1𝐴,𝐴𝐴,𝐴)=−4|Im𝐴|²(Γ(𝐴)(𝐴𝐶+𝐼−Ψ))₈−1|Im𝐴|²Γ(Im𝐴)Ψ(Im𝐴)𝑃. (46) Returning it to complex quaternion form by (ii), (v), and (vii), we have (43).

Theorem 14. Let 𝐴=𝑎+𝑖𝑏,𝐵=𝑐+𝑖𝑑∈𝐇_ℂ,  𝑎,𝑏,𝑐,𝑑∈𝐇, be given with 𝐴∼𝐵. Then (18) has a solution if and only if 𝐴𝐶=𝐶𝐵, (47) in which case the general solution of (18) can be written as 1𝑋=4|Im𝐴|²()1𝐶𝐵−𝐴𝐶+𝑃−|Im𝐴|²(Im𝐴)𝑃(Im𝐵), (48) where 𝑃∈𝐇_ℂ is arbitrary.

Proof. According to (ii) and (v) cases in Proposition 8, (18) is equivalent to [Γ(𝐴)−Ψ(𝐵)]𝑋=𝛿(𝐴,𝐵)𝑋=𝐶. (49) This equation is solvable if and only if 𝛿(𝐴,𝐵)𝛿⁻(𝐴,𝐵)𝐶=𝐶, (50) which is equivalent to Γ(Im𝐴)Ψ(Im𝐵)𝐶=|Im𝐴|²𝐶. (51) Returning it to complex quaternion form by (ii) and (v) cases in Proposition 8, we obtain (Im𝐴)𝐶(Im𝐵)=|Im𝐴|²𝐶, (52) which is equivalent to 𝐶(Im𝐵)=−(Im𝐴)𝐶, and then (47). In that case the general solution of (49) can be expressed as 𝑋=𝛿⁻(𝐴,𝐵)𝐶+2𝐼₈−𝛿⁻(𝐴,𝐵)𝛿(𝐴,𝐵)𝑃, (53) where 𝑃 is an arbitrary vector in 𝐇_ℂ. 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.