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 complex 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 represented 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 complex 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]: Â 1 0 Ã Â i 0 Ã Â 0 1 Ã Â 0 i Ã 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 transpose 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 Â Ã 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. Thus, 2 D C C C det.Q/ 2Q2 2Q2 2Q2 2Q2 D 0 C 1 C 2 C 3 2 Q2 Q2 Q2 Q2 D 0 C 1 C 2 C 3 COMPLEX QUATERNION AND COMPLEX SPLIT QUATERNION MATRICES 49

2 detQ: D .iii/ Let Q Q0E0 Q1E1 Q2E2 Q3E3 and P P0E0 P1E1 P2E2 P3E3 be complexD quaternionC matrices,C QP isC calculated as D C C C

QP .Q0P0 Q1P1 Q2P2 Q3P3/E0 .Q0P1 Q1P0 Q2P3 Q3P2/E1 D C C C .Q0P2 Q2P0 Q1P3 Q3P1/E2 .Q0P3 Q3P0 Q1P2 Q2P1/E3 C C C C C C and from (2.22) we have det.QP / Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 D 0 0 C 0 1 C 0 2 C 0 3 C 1 0 C 1 1 C 1 2 C 1 3 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2: C 2 0 C 2 1 C 2 2 C 2 3 C 3 0 C 3 1 C 3 2 C 3 3 On the other hand, the determinants of Q and P are Q2 Q2 Q2 Q2 and P 2 0 C 1 C 2 C 3 0 C P 2 P 2 P 2, respectively, then 1 C 2 C 3 detQ detP Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 D 0 0 C 0 1 C 0 2 C 0 3 C 1 0 C 1 1 C 1 2 C 1 3 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2: C 2 0 C 2 1 C 2 2 C 2 3 C 3 0 C 3 1 C 3 2 C 3 3 Therefore det.QP / detQ detP: D Additionally, using the complex conjugate and the transpose of a complex quaternion matrix we obtain the determinant of a complex quaternion matrix. t C C t 2 2 2 2 Q Q Q Q Q Q Q Q E0 (2.23) D D 0 C 1 C 2 C 3 and from (2.22) the determinant of Qt QC is 2 det.Qt QC / Q2 Q2 Q2 Q2 (2.24) D 0 C 1 C 2 C 3 so, .detQ/2 det.Qt QC /: (2.25) D If detQ 0; the inverse of a complex quaternion matrix is deﬁned as ¤ 1 Q 1 Q: (2.26) D detQ From (2.12), (2.22) and (2.26) the inverse of a complex quaternion matrix can be written as

1 1 Q .Q0E0 Q1E1 Q2E2 Q3E3/: (2.27) D Q2 Q2 Q2 Q2 0 C 1 C 2 C 3 50 Y. ALAGOZ¨ AND G. OZYURT¨

Example 1. Let Q E0 iE2 E3 be a complex quaternion matrix. Then, the D C C Â 1 2i Ã complex quaternion matrix Q can be written as Q : From (2.27), the D 0 1 inverse of Q is 1 Q E0 iE2 E3 D Â 1 2i Ã : D 0 1 Theorem 2. Complex quaternion matrices satisfy the following properties for Q HC 2 W .i/ E1c cE1;E2c cE2;E3c cE3 for any complex number c; 2 D 2 D D .ii/ Q S det.ImQ/E0 2SQ ImQ; D Q C .iii/ Every complex quaternion matrix Q is expressed as Q Z1 Z2E2 where D C Z1;Z2 M2.C/: 2

Proof. Proofs of .i/ and .iii/ can be easily shown. Now, we will prove .ii/: For C Q Q0E0 Q1E1 Q2E2 Q3E3 H , D C C C 2 2 2 2 2 2 Q .Q Q Q Q /E0 2Q0.Q1E1 Q2E2 Q3E3/ D 0 1 2 3 C C C and using the following equalities 2 2 2 SQ Q0E0; ImQ Q1E1 Q2E2 Q3E3; det.ImQ/ Q Q Q D D C C D 1 C 2 C 3 we get 2 2 Q S det.ImQ/E0 2SQ ImQ: D Q C Theorem 3. For any Q;P HC the following properties are satisﬁed: h t iC 2 .i/ Q Q ; D .ii/ Qt QC ; D C 1 1C .iii/ Q Q if Q is invertible, D 1 1 .iv/ Q Q if Q is invertible, t 1 D 1t .v/ Q Q if Q is invertible, D .vi/ .QP /C QC P C ; D1 1 1 .vii/.QP / P Q if Q and P are invertible. D Proof. Proof of the theorem is easily shown. However, we will prove only .ii/, .iii/ and .vii/: C .ii/ For Q Q0E0 Q1E1 Q2E2 Q3E3 H ; the complex conjugate of a D C C C 2 COMPLEX QUATERNION AND COMPLEX SPLIT QUATERNION MATRICES 51

C complex quaternion matrix is Q Q0E0 Q1E1 Q2E2 Q3E3 and from (2.12) it is obtained that D C C Q Q0E0 Q1E1 Q2E2 Q3E3: D C C The equality (2.15) implies that the transpose of Q HC is 2 t Q Q0E0 Q1E1 Q2E2 Q3E3: D C C Thus, Qt QC : D C .iii/ Let Q Q0E0 Q1E1 Q2E2 Q3E3 H and Q be an invertible complex D C CC C 2 quaternion matrix. We know Q Q0E0 Q1E1 Q2E2 Q3E3: From (2.27) we get D C Á 1 C 1 Q .Q0E0 Q1E1 Q2E2 Q3E3/: D Q2 Q2 Q2 Q2 C C 0 C 1 C 2 C 3 From (2.27) and (2.13), we ﬁnd

1C 1 Q .Q0E0 Q1E1 Q2E2 Q3E3/: D Q2 Q2 Q2 Q2 C C 0 C 1 C 2 C 3 So, Á 1 C QC Q 1 : D .vii/ Let Q Q0E0 Q1E1 Q2E2 Q3E3 and D C C C P P0E0 P1E1 P2E2 P3E3 be invertible complex quaternion matrices. We denoteD QP Cas C C

QP AE0 BE1 CE2 DE3; D C C C for simplicity, where

A Q0P0 Q1P1 Q2P2 Q3P3;B Q0P1 Q1P0 Q2P3 Q3P2; D D C C C Q0P2 Q1P3 Q2P0 Q3P1;D Q0P3 Q1P2 Q2P1 Q3P0: D C C D C C From (2.22) the determinant of QP is written as det.QP / A2 B2 C 2 D2; D C C C where

A2 B2 C 2 D2 C C C Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 D 0 0 C 0 1 C 0 2 C 0 3 C 1 0 C 1 1 C 1 2 C 1 3 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 Q2P 2 C 2 0 C 2 1 C 2 2 C 2 3 C 3 0 C 3 1 C 3 2 C 3 3 52 Y. ALAGOZ¨ AND G. OZYURT¨ and from (2.27) can be found

1 .QP / AE0 BE1 CE2 DE3 .QP / : D det.QP / D A2 B2 C 2 D2 C C C On the other hand, from (2.27), the inverses of P and Q can be written as

1 P0E0 P1E1 P2E2 P3E3 1 Q0E0 Q1E1 Q2E2 Q3E3 P and Q D P 2 P 2 P 2 P 2 D Q2 Q2 Q2 Q2 0 C 1 C 2 C 3 0 C 1 C 2 C 3 and their product is obtained as

AE0 BE1 CE2 DE3 P 1Q 1 : D A2 B2 C 2 D2 C C C Therefore, 1 1 1 .QP / P Q : D C Example 2. Let Q E0 E1;P E0 E2 H . Then, C D C D C 2 C C .i/.QP / E0 E1 E2 E3 E0 E1 E2 E3 P Q D1 1 C ¤ 1 C C D 1 1 .ii/.QP / .E0 E1 E2 E3/ .E0 E1 E2 E3/ Q P D 4 ¤ 4 C D

Example 3. Let Q E0 E1 E2. Then, C D C C C Q Q E0 2E1 2E3 E0 2E1 2E3 Q Q. D C ¤ C C D

With these examples we get the following Corollary for complex quaternion matrices. Corollary 1. Let Q, P HC. Then the followings are satisﬁed: 2 .i/.QP /C P C QC in general; ¤1 1 1 .ii/.QP / Q P in general; C¤ C .iii/ Q Q Q Q in general. ¤

C Deﬁnition 2. For any Q Q0E0 Q1E1 Q2E2 Q3E3 H , .i/ if off-diagonal entries ofDQ are 0 Cthen Q isC called aC diagonal2 matrix and Q is in form of Q Q0E0 Q1E1, t D C .ii/ if Q Q then Q is called a symmetric matrix and Q is in form of Q Q0E0 D D C Q1E1 Q3E3, .iii/ ifCQt Q 1 then Q is called a orthogonal matrix and Q is in form of Q D D Q0E0 Q2E2 and detQ 1; C t D .iv/ if Q Q then Q is called a Hermitian matrix and Q is in form of Q D D Q0E0 Q2E2; C t .v/ if Q Q 1 then Q is called a unitary matrix and Q is in form of Q D D Q0E0 Q1E1 Q3E3 and detQ 1: C C D COMPLEX QUATERNION AND COMPLEX SPLIT QUATERNION MATRICES 53

3. COMPLEX SPLIT QUATERNION MATRICES

A complex split quaternion p is a vector of the form p b0e0 b1e1 b2e2 D C C C b3e3 where b0;b1;b2;b3 are complex numbers. Here e0;e1;e2;e3 denotes the complex split quaternion basis with the below properties f g 2 2 2 e 1; e e e1e2e3 1; (3.1) 1 D 2 D 3 D D e1e2 e2e1 e3; e2e3 e3e2 e1; e3e1 e1e3 e2: (3.2) D D D D D D For details of complex split quaternions, see [3]. A complex split quaternion matrix P is of the form

P P0E0 P1E1 P2E2 P3E3 (3.3) D C C C where P0;P1;P2;P3 are complex numbers. The split quaternion matrix basis E0;E1;E2;E3 satisfy the equalities f g 2 2 2 E E0;E E E0; (3.4) 1 D 2 D 3 D E1E2 E2E1 E3;E2E3 E3E2 E1;E3E1 E1E3 E2: (3.5) D D D D D D These basis elements are 2 2 matrices, [6]: Â 1 0 Ã Â i 0 Ã Â 0 1 Ã Â 0 i Ã E0 ;E1 ;E2 ;E3 : (3.6) D 0 1 D 0 i D 1 0 D i 0 The multiplication rules of the complex 2 2 matrices E0;E1;E2;E3 coincide with the multiplication rules of the complex split quaternion basis elements e0;e1;e2;e3. Hence, there is an isomorphic relation between the vector form and the matrix form of a complex split quaternion. C C Let us denote the algebra of complex split quaternion matrices by HS : HS can be deﬁned with the algebra of 2 2 complex matrices: Â Ã C P0 iP1 P2 iP3 HS P0E0 P1E1 P2E2 P3E3 C C P0;P1;P2;P3 C (3.7) P2 iP3 P0 iP1 D C C C D W 2 C For any P P0E0 P1E1 P2E2 P3E3 H , we deﬁne SP P0E0, the D C C C 2 S D scalar matrix part of P ; ImP P1E1 P2E2 P3E3, the imaginary matrix part of P . The conjugate, the complexD conjugateC andC the total conjugate of a complex split C quaternion matrix are denoted by P;P C ,P respectively these are

P P0E0 P1E1 P2E2 P3E3; (3.8) D C P P0E0 P1E1 P2E2 P3E3 (3.9) D C C C P0E0 P1E1 P2E2 P3E3; D C

C C P .P / P0E0 P1E1 P2E2 P3E3 (3.10) D D 54 Y. ALAGOZ¨ AND G. OZYURT¨

P0E0 P1E1 P2E2 P3E3: D C C C Moreover, for any P P0E0 P1E1 P2E2 P3E3 HS , we can define transpose and adjoint matrix ofDP by P tCand AdjPC andC these are2 t P P0E0 P1E1 P2E2 P3E3; (3.11) D C C AdjP P0E0 P1E1 P2E2 P3E3; (3.12) D AdjP P: (3.13) D The norm of a complex split quaternion matrix Â Ã p11 p12 P P0E0 P1E1 P2E2 P3E3 (3.14) D C C C D p12 p11 is defined as r ˇ 2 2ˇ P ˇ p11 p12 ˇ (3.15) k k D ˇj j j j ˇ where p11 P0 iP1 and p12 P2 iP3. D C D C Definition 3. A determinant of P HC is defined as 2 S

2 2 2 2 detP P detE0 P detE1 P detE2 P detE3: (3.16) D 0 C 1 C 2 C 3 From the determinant of a complex split quaternion basis can be written as detP P 2 P 2 P 2 P 2: (3.17) D 0 C 1 2 3 Theorem 4. For any P;Q HC and C the following properties are satisﬁed: 2 S 2 .i/ detP det.P/ det.P C / detP t ; D D D .ii/ det. P / 2 detP; .iii/ det.PQ/D detP detQ: D

C Proof. .i/ For P P0E0 P1E1 P2E2 P3E3 H , from (3.17) we get D C C C 2 S detP det.P/ det.P C / detP t P 2 P 2 P 2 P 2: D D D D 0 C 1 2 3 C .ii/ For any C and P P0E0 P1E1 P2E2 P3E3 H ; 2 D C C C 2 S P . P0/E0 . P1/E1 . P2/E2 . P3/E3: Thus,D C C C det. P / 2P 2 2P 2 2P 2 2P 2 D 0 C 1 2 3 2 P 2 P 2 P 2 P 2 D 0 C 1 2 3 2 detP: D .iii/ Let P P0E0 P1E1 P2E2 P3E3 and D C C C Q Q0E0 Q1E1 Q2E2 Q3E3 be complex split quaternion matrices, then D C C C PQ COMPLEX QUATERNION AND COMPLEX SPLIT QUATERNION MATRICES 55

.P0Q0 P1Q1 P2Q2 P3Q3/E0 .P0Q1 P1Q0 P2Q3 P3Q2/E1 D C C C C C .P0Q2 P2Q0 P1Q3 P3Q1/E2 .P0Q3 P3Q0 P1Q2 P2Q1/E3 C C C C C C and from (3.17) the determinant of PQ can be found as in the form of det.PQ/ P 2Q2 P 2Q2 P 2Q2 P 2Q2 P 2Q2 P 2Q2 P 2Q2 P 2Q2 D 0 0 C 0 1 C 1 0 C 1 1 C 2 2 C 2 3 C 3 2 C 3 3 P 2Q2 P 2Q2 P 2Q2 P 2Q2 P 2Q2 P 2Q2 P 2Q2 P 2Q2: 0 2 0 3 1 2 1 3 2 0 2 1 3 0 3 1 On the other hand, the determinants of P and Q are P 2 P 2 P 2 P 2 and Q2 0 C 1 2 3 0 C Q2 Q2 Q2 respectively 1 2 3 detP detQ P 2Q2 P 2Q2 P 2Q2 P 2Q2 P 2Q2 P 2Q2 P 2Q2 P 2Q2 D 0 0 C 0 1 C 1 0 C 1 1 C 2 2 C 2 3 C 3 2 C 3 3 P 2Q2 P 2Q2 P 2Q2 P 2Q2 P 2Q2 P 2Q2 P 2Q2 P 2Q2: 0 2 0 3 1 2 1 3 2 0 2 1 3 0 3 1 Thus, det.PQ/ detP detQ: D

Moreover, the determinant of a complex split quaternion matrix can also be found by the complex conjugate and the total conjugate of a complex split quaternion matrix. C C C C 2 2 2 2 P P P P .P P P P /E0 (3.18) D D 0 C 1 2 3 C and from (3.17) the determinant of P C P is found as

C det.P C P / .P 2 P 2 P 2 P 2/2: (3.19) D 0 C 1 2 3 Hence, the determinant of a complex split quaternion matrix can be written as

C .detP /2 det.P C P /: (3.20) D If detP 0; the inverse of a complex split quaternion matrix is deﬁned as ¤ 1 P 1 P: (3.21) D detP From (3.8), (3.17) and (3.21) can be written

1 1 Q .P0E0 P1E1 P2E2 P3E3/: (3.22) D P 2 P 2 P 2 P 2 0 C 1 2 3 56 Y. ALAGOZ¨ AND G. OZYURT¨

1 i Á 1 i Á 1 i Example 4. Let P E0 E1 E2 E3 be a complex split qua- D 2 C 2 2 C 2 Â 1 1 Ã ternion matrix. Then, P can be written as P and from (3.22), the D 0 i inverse of P is calculated as

Â Ã Â Ã 1 1 i 1 i i 1 P C E0 E1 E2 E3 D 2 C 2 C 2 C2 Â 1 i Ã : D 0 i Theorem 5. Complex split quaternion matrices satisfy the following properties for P HC. 2 S .i/ E1c cE1, E2c cE2, E3c cE3 for any complex number c, 2 D 2 D D .ii/ P SP det.ImP /E0 2SP ImP; .iii/ EveryD complex split quaternionC matrix P can be uniquely expressed as P D Z1 Z2E2; where Z1,Z2 M2.C/: C 2 Theorem 6. For any P;Q HC the following properties are satisﬁed: 2 S .i/ detP P 2 ; D1 k k .ii/ P P 1 if P is invertible, D C 1 1C .iii/ P P if P is invertible, h t i 1D h it .iv/ P P 1 if P is invertible, D .v/ .PQ/C P C QC ; D1 1 1 .vi/.PQ/ Q P if P and Q are invertible. D The proof is analogous to the proof of Theorem3.

Example 5. Let P E0 E2 E3 and Q E0 E1. Then, C D C C D CC C .i/.PQ/ E0 E1 2E2 E0 E1 2E3 Q P , CD C ¤ D C .ii/ P.P/ E0 2E1 2E3 E0 2E1 2E3 .P/ P . D C ¤ C C D

With these examples we get the following Corollary for complex split quaternion matrices. Corollary 2. Let P;Q HC: Then the followings are satisﬁed: 2 S .i/ .PQ/C QC P C in general; .ii/ P.P/C¤ .P/C P in general. ¤

C Deﬁnition 4. For any P P0E0 P1E1 P2E2 P3E3 HS , .i/ if off-diagonal entries ofDP are 0Cthen P Cis calledCa diagonal2 matrix and P is in COMPLEX QUATERNION AND COMPLEX SPLIT QUATERNION MATRICES 57 form of P0E0 P1E1, .ii/ if P t PC then P is called a symmetric matrix and P is in form of P D D P0E0 P1E1 P2E2, .iii/ ifC P t CP 1 then P is called an orthogonal matrix and P is in form of D P P0E0 P3E3 and detP 1, D Ct D .iv/ if P P then P is called a Hermitian matrix and P is in form of P D D P0E0 P3E3, C t .v/ if P P 1 then P is called an unitary matrix and P is in form of P D D P0E0 P1E1 P2E2 and detP 1. C C D

4. CONCLUSION This paper has investigated the main properties of complex quaternion and complex split quaternion matrices with the use of 2x2 complex matrix representation of them, respectively. Then, the method of computing the determinant of given complex quaternion and complex split quaternion matrices has been proposed. Although, the determinant properties, conjugate products, special matrices of complex quaternion and complex split quaternion matrices were investigated with the similar methods, but different results were obtained due to the difference of the basis elements of these quaternion matrices.

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Authors’ addresses

Y. Alagoz¨ Yildiz Technical University, Department of Mathematics, 34220 Istanbul, Turkey E-mail address: [email protected] 58 Y. ALAGOZ¨ AND G. OZYURT¨

G. Ozyurt¨ Yildiz Technical University, Department of Mathematics, 34220 Istanbul, Turkey E-mail address: [email protected]