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Some Equivalence Relations and Results Over the Commutative Quaternions and Their Matrices

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Some Equivalence Relations and Results Over the Commutative Quaternions and Their Matrices DOI: 10.1515/auom-2017-0040 An. S¸t. Univ. Ovidius Constant¸a Vol. 25(3),2017, 125{142 Some Equivalence Relations and Results over the Commutative Quaternions and Their Matrices Hidayet H¨udaK¨osaland Murat Tosun Abstract In this paper, we give some equivalence relations and results over the commutative quaternions and their matrices. In this sense, con- similarity, semisimilarity, and consemisimilarity over the commutative quaternion algebra and commutative quaternion matrix algebra are es- tablished. Equalities of these equivalence relations are explicitly de- termined. Also Syvester-s-Conjugate commutative quaternion matrix equations are studied by means of real representation of the commu- tative quaternion matrices and consimilarity of the two commutative quaternion matrices. 1 Introduction In 1843, Hamilton introduced the concept of real quaternions, which is defined by [1] K = fq = q0 + q1i + q2j + q3k : q0; q1; q2; q3 2 R and i; j; k2 = Rg ; where i2 = j2 = k2 = −1; ij = −ji = k; ik = −ki = −j; jk = −kj = i: Key Words: Commutative quaternion, equivalence relation, coneigenvalue, Slyvester conjugate matrix equation. 2010 Mathematics Subject Classification: Primary 15B33; Secondary 15A24. Received: 20.11.2016 Revised: 15.12.2016 Accepted: 21.12.2016 125 Some Equivalence Relations and Results over the Commutative Quaternions and Their Matrices 126 The real quaternion algebra plays an important role in quantum physics, kine- matic, differential geometry, game development, image processing and signal processing, etc. Real quaternions are a naturel extension of the complex num- bers which are also the extension of the real numbers. The multiplication of real quaternions is non-commutative. Thus, all results about complex numbers cannot be generalized in real quaternions. There are a lot of works associated with real quaternions. For instance, Tian defined two types of universal factor- ization equalities for real quaternions and gave solutions of ax−xb = c in K, [2]. As well as the similarity and consimilarity of elements of the real quaternion, octanion, and sedenion algebras, Tian considered the similarity and consimi- larity of general real Cayley-Dickson algebras in [3]. Also, the author studied the solutions of two fundamental equations, ax = xb and ax = xb; by means of similarity and consimilarity relations. In [4], Tian defined semisimilarity and consemisimilarity of real quaternions and investigated the general solutions of systems xay = b; ybx = a; and xay = b; ybx = a in K: One of the applica- tions of real quaternions is also the quaternion matrix theory. In [5], Baker investigated the right eigenvalues of the quaternion matrices using the topo- logical approach. Besides, Huang and So studied on the left eigenvalues of real quaternion matrices [6]. Huang discussed the consimilarity of the quaternion matrices and obtained their Jordan canonical form, using the consimilarity [7]. Jiang and Wei studied the Kalman-Yakubovich-Conjugate matrix equa- tion ;X − AXBe = C; in K (where Xe = −jXj ) via the real representation of real quaternion matrices [8]. Moreover, Jiang and Ling studied the prob- lem of the solution of the Sylvester-Conjugate real quaternion matrix equation ;AXe −XB = C; by means of real representation of the real quaternions matrix [9]. After the introduction of real quaternions, the set of commutative quaternions was first introduced by Segre [10]. This number system is sometimes called the system of reduced bi-quaternion. The set of commutative quaternions is four-dimensional like the set of real quaternions. However this set con- tains zero-divisor elements. Commutative quaternions are extensively stud- ied and applied to several problems in various areas. Catoni et al. studied the functions of commutative quaternion variable and obtained generalized Cauchy-Riemann conditions [11]. In [12], the authors introduced digital sig- nal and image processing, using commutative quaternions. Also, they dis- cussed the efficient algorithms of the discrete commutative quaternion Fourier transform, convolution, correlation, and phase-only correlation. In [13], the authors developed the algorithms for calculating the eigenvalues-eigenvectors and the singular value decomposition of commutative quaternion matrices. Moreover, they represented the color images in the digital media by commu- tative quaternion matrices and applied the techniques, such as separation, Some Equivalence Relations and Results over the Commutative Quaternions and Their Matrices 127 compression, image enhancement and denosing, to the color images by us- ing these matrices. In [14], the authors investigated two types of multistate Hopfield neural networks based on commutative quaternions. In [15], the au- thors defined commutative quaternion canonical transform which is the gen- eralization of commutative quaternion Fourier transform. Also, Kosal and Tosun investigated some algebraic properties of commutative quaternion ma- trices by means of complex representation of commutative quatrnion matrices [16]. In [17], Kosal et al. constructed , by means of real representation of a real matrix, some explicit expression of the solutions of the matrix equations, X − AXB = C; X − AXBe = C; and X − AXBe = C; which, for convenience, are called the Kalman-Yakubovich-Conjugate commutative quaternion matrix equations. This article is organized as follows. In Section 2, after we give algebraic proper- ties of commutative quaternions, consimilarity, semisimilarity and consemisim- ilarity of commutative quaternions are defined. Also, equalities of these equiva- lence relations are studied. In Section 3, we define consimilarity, semisimilarity, and consemisimilarity of commutative quaternion matrices and obtain equal- ities of these equivalence relations. Lastly, we introduce Syvester-s-Conjugate matrix equations, A sX − XB = C,(s = 1; 2; 3) via real representation of commutative quaternion matrices and consimilarity of the two commutative quaternion matrices. Throughout this paper, the following notations are used. Let R; K; and H denote the real numbers field, the real quaternion skew field, and the commutative quaternion ring, respectively. Rm×n (Km×n or Hm×n) denotes the set of all matrices on R (K or H) : 2 Equivalence Relations and Results over the Commu- tative Quaternions The set of commutative quaternions is expressed by H = fa = a0 + a1i + a2j + a3k : a0; a1; a2; a3 2 R and i; j; k2 = R g ; (1) where i2 = k2 = −1; j2 = 1; ij = ji = k; jk = kj = i; ki = ik = −j: It is clear that multiplication in H is commutative. Summation of the com- mutative quaternions a = a0 + a1i + a2j + a3k; b = b0 + b1i + b2j + b3k 2 H is defined as a + b = (a0 + b0) + (a1 + b1) i + (a2 + b2) j + (a3 + b3) k. Scalar multiplication of a commutative quaternion a 2 H with a scalar λ 2 R is defined as λa = λ (a0 + a1i + a2j + a3k) = λa0 + λa1i + λa2j + λa3k. In ad- dition, quaternionic multiplication of two commutative quaternions a; b 2 H Some Equivalence Relations and Results over the Commutative Quaternions and Their Matrices 128 is defined ab = (a0b0 − a1b1 + a2b2 − a3b3) + (a1b0 + a0b1 + a3b2 + a2b3) i + (a0b2 + a2b0 − a1b3 − a3b1) j + (a3b0 + a0b3 + a1b2 + a2b1) k: 1 There are three types of conjugates of a 2 H. They are a = a0 − a1i + a2j − 2 3 a3k; a = a0 + a1i − a2j − a3k; and a = a0 − a1i − a2j + a3k: Also, the norm of a 2 H is defined as q p4 1 2 3 4 2 2 2 2 kak = a ( a)( a)( a) = (a0 + a2) + (a1 + a3) (a0 − a2) + (a1 − a3) : (2) In case of a0 + a2 = 0; a1 + a3 = 0 or a0 − a2 = 0; a1 − a3 = 0 (3) norm of a is equal to zero. The planes of equations (3) are called planes of the zero divisors or characteristic planes [11]. If a 2 H and kak 6= 0 then a has multiplicative inverse. Multiplicative inverse (1a)(2a)(3a) of a is equal to a−1 = [11]. kak4 It is nearby to identify a commutative quaternion a 2 H with a real vector a 2 R4: We may define any commutative quaternion as 0 1 a0 ∼ B a1 C a = a0 + a1i + a2j + a3k = a = B C : @ a2 A a3 Then multiplication of a and b can be shown, with the help of ordinary matrix multiplication, 0 1 0 1 a0 −a1 a2 −a3 b0 B a1 a0 a3 a2 C B b1 C ab = ba =∼ B C B C = ' (a) b; (4) @ a2 −a3 a0 −a1 A @ b2 A a3 a2 a1 a0 b3 where ' (a) is sometimes called fundamental matrix of a [11]. Theorem 1. ([12]) Let a; b 2 H and λ 2 R. Then the following identities hold: 1. ' (ab) = ' (a) ' (b) ; 2. ' (' (a) b) = ' (a) ' (b) ; 3. a = b , ' (a) = ' (b) ; 4. ' (a + b) = ' (a) + ' (b) ; 5. ' (λa) = λϕ (a) ; 1 2 3 6. trace (' (a)) = a + a + a + a = 4a0; 7. kak4 = jdet (' (a))j : Some Equivalence Relations and Results over the Commutative Quaternions and Their Matrices 129 Definition 1. Two commutative quaternions a and b said to be consimilar according to sth (s = 1; 2; 3) conjugate if there exists a commutative quater- c nion p; kpk 6= 0 such that spap−1 = b; this is written as a ∼s b: Consimilarity c relation ∼ is an equivalence relation on the commutative quaternions. Theorem 2. Let a; b 2 H and a is consimilar to b according to sth (s = 1; 2; 3) conjugate. Then norm of a is equal to norm of b. Proof. If a and b are consimilar according to sth (s = 1; 2; 3) conjugate, then there exists p such as spap−1 = b: In last equation, we can calculate the norm both sides. Then we have s −1 k pk kak p = kbk : −1 Since kspk = kpk and p−1 = kpk , we get kak = kbk : Definition 2. Two commutative quaternions a and b are said to be semisim- ilar if there exist a commutative quaternions x and y that satisfying the equa- tion xay = b and ybx = a; (5) this is written as a ≈ b: Semisimilarity relation ≈ is an equivalence relation on H: Theorem 3.
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