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Some Properties of Complex Quaternion and Complex Split Quaternion Matrices

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Some Properties of Complex Quaternion and Complex Split Quaternion Matrices Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 20 (2019), No. 1, pp. 45–58 DOI: 10.18514/MMN.2019.2550 SOME PROPERTIES OF COMPLEX QUATERNION AND COMPLEX SPLIT QUATERNION MATRICES Y. ALAGOZ¨ AND G. OZYURT¨ Received 05 March, 2018 Abstract. The aim of this study is to investigate some properties of complex quaternion and com- plex split quaternion matrices. To verify this, we use 2x2 complex matrix representation of these quaternions. Moreover, we present a method to find the determinant of complex quaternion and complex split quaternion matrices. Finally, we research some special matrices for quaternions above. 2010 Mathematics Subject Classification: 15B33; 11R52 Keywords: complex quaternion matrix, complex split quaternion matrix 1. INTRODUCTION The real quaternion algebra H is a four dimensional vector space over the real 4 number field R and e0;e1;e2;e3 denote the basis of H and basis of R . The set of real quaternions are a number system that extends the complex numbers field C. Irish mathematician Sir William Rowan Hamilton introduced it in 1843, which is represented as H a a0e0 a1e1 a2e2 a3e3 a0;a1;a2;a3 R Df D C C C W 2 g 2 2 2 where e0 acts an identity and e1 e2 e3 e1e2e3 1. Since e2e3 e3e2 it is obvious that the real quaternionsD areD noncommutativeD D and differ from complex¤ numbers and real numbers. Furthermore any real quaternion can be respesented by a 2 2 complex matrix, [2]. A complex quaternion it is called also biquaternion q can be written as q a0e0 a1e1 a2e2 a3e3 where a0;a1;a2;a3 C and its D C C C 2 basis elements e0;e1;e2;e3 satisfy the real quaternion multiplication rules. In [5] and [7] conjugates, 2 2 complex matrices corresponding to basis elements of complex quaternions are expressed. In 1849, James Cockle introduced the set of real split quaternions which is repres- ented as HS p b0e0 b1e1 b2e2 b3e3 b0;b1;b2;b3 R Df D C C C W 2 g c 2019 Miskolc University Press 46 Y. ALAGOZ¨ AND G. OZYURT¨ 2 2 2 where e1 1; e2 e3 1 and e1e2e3 1: Real split quaternions are noncommut- ative, too,D [6 ]. Also,D anyD real split quaternionsD can be represented by 2 2 complex matrix, [1]. While coefficients of a real split quaternion are complex numbers, then it is called complex split quaternion. The basis elements of a complex split quaternion have the same rules of a real split quaternion multiplication, [3]. In this study, firstly we associated the results we obtained from the conjugates of the complex quaternion with the real quaternions. Also, we give some properties of matrix representation of complex quaternions and complex split quaternions by expressing these quaternions as 2 2 complex matrices .M2 .C// with using matrices corresponding to the basis of complex quaternions and complex split quaternions. Moreover, we obtain a method to find the determinant for these form of quaternions. Finally, we investigate some special matrices for complex quaternion and complex split quaternion matrices. 2. COMPLEX QUATERNION MATRICES A real quaternion a is a vector of the form a a0e0 a1e1 a2e2 a3e3 D C C C where a0;a1;a2;a3 are real numbers. Here e0;e1;e2;e3 denotes the set of real quaternion basis with the properties f g 2 2 2 e e e e1e2e3 1; (2.1) 1 D 2 D 3 D D e1e2 e2e1 e3; e2e3 e3e2 e1; e3e1 e1e3 e2: (2.2) D D D D D D A real quaternion a can be written as a Sa Va where Sa a0e0 is the scalar D C D part and Va a1e1 a2e2 a3e3 is the vector part of a. For any real quaternion D C C a a0e0 a1e1 a2e2 a3e3; the conjugate of a is a a0e0 a1e1 a2e2 a3e3 D C C C q D and the norm of a is a paa paa a2 a2 a2 a2. For details, see k k D D D 0 C 1 C 2 C 3 [2]. A complex quaternion q is of the form q A0e0 A1e1 A2e2 A3e3 where D C C C A0;A1;A2;A3 are complex numbers and the elements of e0;e1;e2;e3 multiply as f P g in real quaternions. Also, a complex quaternion q can be written as .ak ibk/ek where a ;b are real numbers for 0 k 3. Here i denotes the complexC unit and k k Ä Ä commutes with e0;e1;e2;e3: For any complex quaternion q A0e0 A1e1 A2e2 A3e3; the quaternion con- D C C C 2 2 2 2 jugate of q is q A0e0 A1e1 A2e2 A3e3 and qq qq A0 A1 A2 A3: D c D D C C C The complex conjugate of q is q A0e0 A1e1 A2e2 A3e3 and the Hermitian c D C C C conjugate of q is .q/ A0e0 A1e1 A2e2 A3e3: For more information of com- plex quaternions the readerD is referred to [5] and [4]. For a complex quaternion q .a0 ib0/e0 .a1 ib1/e1 .a2 ib2/e2 .a3 ib3/e3; we express the equalitiesD C below relatedC toC real quaternionsC C and complexC quaternionsC with using the complex conjugate and the Hermitian conjugate of a complex quaternion. c 2 2 q q a b 2i .Va V / (2.3) D C C b COMPLEX QUATERNION AND COMPLEX SPLIT QUATERNION MATRICES 47 c 2 2 qq a b 2i .Va V / (2.4) D C b c 2 2 .q/ q a b 2i .SaV S Va Va V / (2.5) D k k C k k C b b b c 2 2 q .q/ a b 2i .SaV S Va Va V / (2.6) D k k C k k C b b C b where a a0e0 a1e1 a2e2 a3e3, b b0e0 b1e1 b2e2 b3e3 and denotes the vectorD productC in R3C: C D C C C A complex quaternion matrix Q is of the form Q Q0E0 Q1E1 Q2E2 Q3E3 (2.7) D C C C where Q0;Q1;Q2;Q3 are complex numbers. The complex quaternion matrix basis E0;E1;E2;E3 satisfying the equalities f g 2 2 2 E E E E0; (2.8) 1 D 2 D 3 D E1E2 E2E1 E3;E2E3 E3E2 E1;E3E1 E1E3 E2: (2.9) D D D D D D These basis elements are 2 2 matrices, [5]:  1 0 à  i 0 à  0 1 à  0 i à E0 ;E1 ;E2 ;E3 : (2.10) D 0 1 D 0 i D 1 0 D i 0 The multiplication rules of the 2 2 complex matrices E0;E1;E2;E3 satisfy the multiplication rules of the complex quaternion basis elements e0;e1;e2;e3. Hence, there is an isomorphic relation between the vector form and the matrix form of a complex quaternion. We denote the algebra of complex quaternion matrices by HC and define with the algebra of 2 2 complex matrices: ( ! ) C Q0 iQ1 Q2 iQ3 H Q0E0 Q1E1 Q2E2 Q3E3 C C Q0;Q1;Q2;Q3 C (2.11) D C C C D Q2 iQ3 Q0 iQ1 W 2 C C For any Q Q0E0 Q1E1 Q2E2 Q3E3 H ; we define SQ Q0E0, the D C C C 2 D scalar matrix part of Q; ImQ Q1E1 Q2E2 Q3E3, the imaginary matrix part of Q. The conjugate, the complexD conjugateC andC the total conjugate of a complex C quaternion matrix are denoted by Q;QC ,Q respectively these are Q Q0E0 Q1E1 Q2E2 Q3E3; (2.12) D C Q Q0E0 Q1E1 Q2E2 Q3E3 (2.13) D C C C Q0E0 Q1E1 Q2E2 Q3E3; D C 48 Y. ALAGOZ¨ AND G. OZYURT¨ C C Q .Q / Q0E0 Q1E1 Q2E2 Q3E3 (2.14) D D Q0E0 Q1E1 Q2E2 Q3E3: D C C C In addition, for any Q Q0E0 Q1E1 Q2E2 Q3E3 H , we can define trans- pose and adjoint matrixD of Q byCQt andCAdjQ respectivelyC 2 write down as t Q Q0E0 Q1E1 Q2E2 Q3E3; (2.15) D C C AdjQ Q0E0 Q1E1 Q2E2 Q3E3: (2.16) D So we can get AdjQ Q; (2.17) D t QC Q : (2.18) D The norm of a complex quaternion matrix  à q11 q12 Q Q0E0 Q1E1 Q2E2 Q3E3 (2.19) D C C C D q q 12 11 is defined as r ˇ 2 2ˇ Q ˇ q11 q12 ˇ (2.20) k k D ˇj j C j j ˇ where q11 Q0 iQ1 and q12 Q2 iQ3. D C D C Definition 1. A determinant of Q HC is defined as 2 2 2 2 2 detQ Q detE0 Q detE1 Q detE2 Q detE3: (2.21) D 0 C 1 C 2 C 3 Using the determinant of a complex quaternion matrix basis the above determinant can be written as detQ Q2 Q2 Q2 Q2: (2.22) D 0 C 1 C 2 C 3 Theorem 1. For any Q;P HC and C the following properties are satisfied: .i/ detQ det.Q/ det.QC2/ detQt 2; D D D .ii/ det.Q/ 2 detQ; .iii/ det.QPD/ detQ detP: D C Proof. .i/ For Q Q0E0 Q1E1 Q2E2 Q3E3 H , from (2.22) it can be found easily that D C C C 2 detQ det.Q/ det.QC / detQt Q2 Q2 Q2 Q2: D D D D 0 C 1 C 2 C 3 .ii/ For any C; we have Q .Q0/E0 .Q1/E1 .Q2/E2 .Q3/E3.
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