Moreover, we present a method to find the determinant 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. ALAG ¨OZ 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.

2010Mathematics 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 number field R and e₀; e₁; e₂; e₃ denote the basis of H and basis of R⁴. The set of real quaternions are a number system that extends the complex numbers fieldC.

Irish mathematician Sir William Rowan Hamilton introduced it in 1843, which is represented as

HD faDa0e0Ca1e1Ca2e2Ca3e3Wa0; a1; a2; a32Rg

where e0 acts an identity and e₁² De₂² De₃² De1e2e3D 1. Since e2e3¤e3e2

it is obvious that the real quaternions are noncommutative and differ from complex numbers and real numbers. Furthermore any real quaternion can be respesented by a 22 complex matrix, [2]. A complex quaternion it is called also biquaternion qcan be written asqDa0e0Ca1e1Ca2e2Ca3e3wherea0; a1; a2; a32Cand its basis elementse0; e1; e2; e3satisfy the real quaternion multiplication rules. In [5] and [7] conjugates,22complex 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

H_SD fpDb0e0Cb1e1Cb2e2Cb3e3Wb0; b1; b2; b32Rg

c 2019 Miskolc University Press

wheree₁²D 1; e₂²De₃²D1ande1e2e3D1:Real split quaternions are noncommut- ative, too, [6]. Also, any real split quaternions can be represented by22complex 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 as22complex 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 quaternionais a vector of the formaDa0e0Ca1e1Ca2e2Ca3e3

where a0; a1; a2; a3 are real numbers. Here fe0; e1; e2; e3gdenotes the set of real quaternion basis with the properties

e₁²De₂²De²₃De1e2e3D 1; (2.1) e1e2D e2e1De3; e2e3D e3e2De1; e3e1D e1e3De2: (2.2) A real quaternionacan be written asaDSaCVawhereSaDa0e0is the scalar part and Va Da1e1Ca2e2Ca3e3 is the vector part ofa. For any real quaternion aDa0e0Ca1e1Ca2e2Ca3e3;the conjugate ofaisaDa0e0 a1e1 a2e2 a3e3

and the norm ofais kak Dp

aaDp aaD

q

a²₀Ca²₁Ca²₂Ca²₃. For details, see [2].

A complex quaternionq is of the formqDA0e0CA1e1CA2e2CA3e3 where A0; A1; A2; A3are complex numbers and the elements offe0; e1; e2; e3gmultiply as in real quaternions. Also, a complex quaternionqcan be written asP

.a_kCi b_k/ e_k wherea_k; b_k are real numbers for0k3. Herei denotes the complex unit and commutes withe0; e1; e2; e3:

For any complex quaternionqDA0e0CA1e1CA2e2CA3e3;the quaternion con- jugate ofqisqDA₀e₀ A₁e₁ A₂e₂ A₃e₃andqqDqqDA²₀CA²₁CA²₂CA²₃: The complex conjugate ofqisq^cDA0e0CA1e1CA2e2CA3e3and the Hermitian conjugate ofqis.q/^cDA0e0 A1e1 A2e2 A3e3:For more information of com- plex quaternions the reader is referred to [5] and [4]. For a complex quaternion q D.a0Ci b0/ e0C.a1Ci b1/ e1C.a2Ci b2/ e2C.a3Ci b3/ e3; we express the equalities below related to real quaternions and complex quaternions with using the complex conjugate and the Hermitian conjugate of a complex quaternion.

q^cqDa²Cb²C2i .VaVb/ (2.3)

qq^cDa²Cb² 2i .VaVb/ (2.4) .q/^cqD kak²C kbk²C2i .SaV_b S_bVa VaV_b/ (2.5) q .q/^c D kak²C kbk²C2i .SaV_b S_bVaCVaV_b/ (2.6) whereaDa₀e₀Ca₁e₁Ca₂e₂Ca₃e₃,bDb₀e₀Cb₁e₁Cb₂e₂Cb₃e₃anddenotes the vector product inR³:

A complex quaternion matrixQis of the form

QDQ0E0CQ1E1CQ2E2CQ3E3 (2.7) whereQ0; Q1; Q2; Q3are complex numbers. The complex quaternion matrix basis fE0; E1; E2; E3gsatisfying the equalities

E₁²DE₂²DE₃²D E0; (2.8) E1E2D E2E1DE3; E2E3D E3E2DE1; E3E1D E1E3DE2: (2.9) These basis elements are22matrices, [5]:

E0D

1 0 0 1

; E1D

i 0 0 i

; E2D

0 1 1 0

; E3D 0 i

i 0

: (2.10) The multiplication rules of the 22 complex matrices E0; E1; E2; E3 satisfy the multiplication rules of the complex quaternion basis elementse0; 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 byH^Cand define with the algebra of22complex matrices:

H^CD (

Q0E0CQ1E1CQ2E2CQ3E3D Q0CiQ1 Q2CiQ3

Q2CiQ3 Q0 iQ1

!

WQ0;Q1;Q2;Q32C )

(2.11) For anyQDQ0E0CQ1E1CQ2E2CQ3E32H^C;we defineSQDQ0E0, the scalar matrix part ofQ; ImQDQ1E1CQ2E2CQ3E3, the imaginary matrix part ofQ. The conjugate, the complex conjugate and the total conjugate of a complex quaternion matrix are denoted byQ; Q^C, QC

respectively these are

QDQ0E0 Q1E1 Q2E2 Q3E3; (2.12) Q^C DQ0E0CQ1E1CQ2E2CQ3E3 (2.13)

DQ0E0 Q1E1CQ2E2 Q3E3;

QC

D.Q^C/DQ0E0 Q1E1 Q2E2 Q3E3 (2.14) DQ0E0CQ1E1 Q2E2CQ3E3:

In addition, for anyQDQ0E0CQ1E1CQ2E2CQ3E32H^C, we can define trans- pose and adjoint matrix ofQbyQ^t andAdjQrespectively write down as

Q^t DQ0E0CQ1E1 Q2E2CQ3E3; (2.15) AdjQDQ0E0 Q1E1 Q2E2 Q3E3: (2.16) So we can get

AdjQDQ; (2.17)

Q^C D Qt

: (2.18)

The norm of a complex quaternion matrix

QDQ0E0CQ1E1CQ2E2CQ3E3D

q11 q12

q₁₂ q₁₁

(2.19) is defined as

kQk D rˇ

ˇˇjq11j²C jq12j² ˇ ˇ

ˇ (2.20)

whereq11DQ0CiQ1andq12DQ2CiQ3.

Definition 1. A determinant ofQ2H^Cis defined as

detQDQ²₀detE0CQ₁²detE1CQ²₂detE2CQ²₃detE3: (2.21) Using the determinant of a complex quaternion matrix basis the above determinant can be written as

detQDQ²₀CQ²₁CQ₂²CQ²₃: (2.22) Theorem 1. For anyQ; P 2H^Cand2Cthe following properties are satisfied:

.i /detQDdet.Q/Ddet.Q^C/Ddet Q^t

; .i i /det.Q/D²detQ;

.i i i /det.QP /DdetQdetP:

Proof. .i /ForQDQ0E0CQ1E1CQ2E2CQ3E32H^C, from (2.22) it can be found easily that

detQDdet.Q/Ddet.Q^C/Ddet Q^t

DQ²₀CQ²₁CQ₂²CQ²₃:

.i i / For any2C;we have QD.Q0/ E0C.Q1/ E1C.Q2/ E2C.Q3/ E3. Thus,

det.Q/D²Q₀²C²Q₁²C²Q₂²C²Q₃² D² Q²₀CQ₁²CQ²₂CQ₃²

D²detQ:

.i i i /LetQDQ0E0CQ1E1CQ2E2CQ3E3andP DP0E0CP1E1CP2E2CP3E3

be complex quaternion matrices,QP is calculated as

QP D.Q0P0 Q1P1 Q2P2 Q3P3/ E0C.Q0P1CQ1P0CQ2P3 Q3P2/ E1

C.Q0P2CQ2P0 Q1P3CQ3P1/ E2C.Q0P3CQ3P0CQ1P2 Q2P1/ E3

and from (2.22) we have

det.QP /DQ²₀P₀²CQ²₀P₁²CQ₀²P₂²CQ₀²P₃²CQ₁²P₀²CQ²₁P₁²CQ²₁P₂²CQ²₁P₃² CQ₂²P₀²CQ₂²P₁²CQ²₂P₂²CQ²₂P₃²CQ²₃P₀²CQ²₃P₁²CQ²₃P₂²CQ²₃P₃²: On the other hand, the determinants ofQandP areQ²₀CQ²₁CQ₂²CQ₃²andP₀²C P₁²CP₂²CP₃², respectively, then

detQdetP

DQ²₀P₀²CQ₀²P₁²CQ₀²P₂²CQ₀²P₃²CQ₁²P₀²CQ²₁P₁²CQ²₁P₂²CQ²₁P₃² CQ₂²P₀²CQ₂²P₁²CQ²₂P₂²CQ²₂P₃²CQ²₃P₀²CQ²₃P₁²CQ²₃P₂²CQ²₃P₃²: Therefore

det.QP /DdetQdetP:

Additionally, using the complex conjugate and the transpose of a complex qua- ternion matrix we obtain the determinant of a complex quaternion matrix.

Q^tQ^C DQ^CQ^tD Q²₀CQ₁²CQ²₂CQ²₃

E0 (2.23)

and from (2.22) the determinant ofQ^tQ^C is

det.Q^tQ^C/D Q²₀CQ₁²CQ²₂CQ²₃2

(2.24) so,

.detQ/²Ddet.Q^tQ^C/: (2.25)

If detQ¤0;the inverse of a complex quaternion matrix is defined as Q ¹D 1

detQQ: (2.26)

From (2.12), (2.22) and (2.26) the inverse of a complex quaternion matrix can be written as

Q ¹D 1

Q²₀CQ²₁CQ₂²CQ²₃.Q0E0 Q1E1 Q2E2 Q3E3/ : (2.27)

Example 1. LetQDE0CiE2CE3 be a complex quaternion matrix. Then, the complex quaternion matrixQ can be written asQD

1 2i 0 1

:From (2.27), the inverse ofQis

Q ¹DE0 iE2 E3

D

1 2i 0 1

:

Theorem 2. Complex quaternion matrices satisfy the following properties for Q2H^CW

.i / E1cDcE1; E2cDcE2; E3cDcE3for any complex numberc;

.i i / Q²DS_Q² det.ImQ/ E0C2SQImQ;

.i i i /Every complex quaternion matrix Q is expressed asQDZ1CZ2E2 where Z1;Z22M2.C/:

Proof. Proofs of.i /and.i i i /can be easily shown. Now, we will prove.i i /:For QDQ0E0CQ1E1CQ2E2CQ3E32H^C,

Q²D.Q₀² Q²₁ Q²₂ Q²₃/E0C2Q0.Q1E1CQ2E2CQ3E3/ and using the following equalities

SQDQ0E0; ImQDQ1E1CQ2E2CQ3E3; det.ImQ/DQ²₁CQ²₂CQ₃² we get

Q²DS_Q² det.ImQ/ E0C2SQImQ:

Theorem 3. For anyQ; P 2H^Cthe following properties are satisfied:

.i / QDh QtiC

; .i i / Q^tD Q^C

; .i i i / Q^C 1

D Q ¹C

ifQis invertible, .iv/ Q ¹

D Q ¹

ifQis invertible, .v/ Q^t 1

D Q ¹t

ifQis invertible, .vi / .QP /^C DQ^CP^C;

.vi i / .QP / ¹DP ¹Q ¹ifQandP are invertible.

Proof. Proof of the theorem is easily shown. However, we will prove only.i i /, .i i i /and.vi i /:

.i i / For QDQ0E0CQ1E1CQ2E2CQ3E32H^C; the complex conjugate of a

complex quaternion matrix isQ^C DQ0E0 Q1E1CQ2E2 Q3E3and from (2.12) it is obtained that

Q^C

DQ0E0CQ1E1 Q2E2CQ3E3: The equality (2.15) implies that the transpose ofQ2H^Cis

Q^t DQ0E0CQ1E1 Q2E2CQ3E3: Thus,

Q^t D Q^C :

.i i i /LetQDQ0E0CQ1E1CQ2E2CQ3E32H^CandQbe an invertible complex quaternion matrix. We knowQ^C DQ0E0 Q1E1CQ2E2 Q3E3:From (2.27) we get

Q^C 1

D 1

Q²₀CQ₁²CQ²₂CQ²₃.Q0E0CQ1E1 Q2E2CQ3E3/ : From (2.27) and (2.13), we find

Q ^1C

D 1

Q²₀CQ²₁CQ²₂CQ₃².Q0E0CQ1E1 Q2E2CQ3E3/ : So,

Q^C 1

D Q ^1C : .vi i /LetQDQ0E0CQ1E1CQ2E2CQ3E3and

P DP0E0CP1E1CP2E2CP3E3 be invertible complex quaternion matrices. We denoteQP as

QP DAE0CBE1CCE2CDE3; for simplicity, where

ADQ0P0 Q1P1 Q2P2 Q3P3; BDQ0P1CQ1P0CQ2P3 Q3P2; C DQ0P2 Q1P3CQ2P0CQ3P1; DDQ0P3CQ1P2 Q2P1CQ3P0: From (2.22) the determinant ofQP is written as

det.QP /DA²CB²CC²CD²; where

A²CB²CC²CD²

DQ²₀P₀²CQ₀²P₁²CQ₀²P₂²CQ₀²P₃²CQ²₁P₀²CQ²₁P₁²CQ²₁P₂²CQ²₁P₃² CQ²₂P₀²CQ²₂P₁²CQ²₂P₂²CQ²₂P₃²CQ²₃P₀²CQ₃²P₁²CQ₃²P₂²CQ₃²P₃²

and from (2.27) can be found .QP / ¹D .QP /

det.QP / DAE0 BE1 CE2 DE3

A²CB²CC²CD² : On the other hand, from (2.27), the inverses ofP andQcan be written as

P ¹DP0E0 P1E1 P2E2 P3E3

P₀²CP₁²CP₂²CP₃² andQ ¹DQ0E0 Q1E1 Q2E2 Q3E3

Q₀²CQ²₁CQ²₂CQ₃² and their product is obtained as

P ¹Q ¹DAE0 BE1 CE2 DE3

A²CB²CC²CD² : Therefore,

.QP / ¹DP ¹Q ¹:

Example2. LetQDE₀CE₁; P DE₀CE₂2H^C. Then,

.i / .QP /^C DE0 E1CE2 E3¤E0 E1CE2CE3DP^CQ^C

.i i / .QP / ¹D¹₄.E0 E1 E2 E3/¤¹₄.E0 E1 E2CE3/DQ ¹P ¹ Example3. LetQDE0CE1CE2. Then,

Q QC

DE0C2E1 2E3¤E0C2E1C2E3D QC

Q.

With these examples we get the following Corollary for complex quaternion matrices.

Corollary 1. LetQ,P 2H^C.Then the followings are satisfied:

.i / .QP /^C¤P^CQ^C in general;

.i i / .QP / ¹¤Q ¹P ¹in general;

.i i i / Q QC

¤ QC

Qin general.

Definition 2. For anyQDQ0E0CQ1E1CQ2E2CQ3E32H^C,

.i /if off-diagonal entries ofQare0thenQ is calleda diagonal matrixandQis in form ofQDQ0E0CQ1E1,

.i i /ifQ^tDQthenQis calleda symmetric matrix andQis in form ofQDQ0E0C Q1E1CQ3E3,

.i i i /ifQ^t DQ ¹thenQ is calleda orthogonal matrixandQis in form ofQD Q0E0CQ2E2and detQD1;

.iv/if Qt

DQ then Q is called a Hermitian matrixand Q is in form of QD Q0E0CQ2E2;

.v/ if Qt

DQ ¹ then Q is called a unitary matrix and Q is in form of QD Q0E0CQ1E1CQ3E3and detQD1:

3. COMPLEX SPLIT QUATERNION MATRICES

A complex split quaternionpis a vector of the formpDb0e0Cb1e1Cb2e2C b3e3whereb0; b1; b2; b3are complex numbers. Herefe0; e1;e2; e3gdenotes the com- plex split quaternion basis with the below properties

e²₁D 1; e₂²De₃²De1e2e3D1; (3.1) e1e2D e2e1De3; e2e3D e3e2D e1; e3e1D e1e3De2: (3.2) For details of complex split quaternions, see [3].

A complex split quaternion matrixP is of the form

P DP0E0CP1E1CP2E2CP3E3 (3.3) whereP0; P1; P2; P3are complex numbers. The split quaternion matrix basis fE0; E1; E2; E3gsatisfy the equalities

E₁²D E0; E₂²DE₃²DE0; (3.4) E1E2D E2E1DE3; E2E3D E3E2D E1; E3E1D E1E3DE2: (3.5) These basis elements are22matrices, [6]:

E0D

1 0 0 1

; E1D

i 0 0 i

; E2D

0 1 1 0

; E3D

0 i i 0

: (3.6) The multiplication rules of the complex22matricesE0; E1; E2; E3coincide with the multiplication rules of the complex split quaternion basis elementse0; e1;e2; e3. Hence, there is an isomorphic relation between the vector form and the matrix form of a complex split quaternion.

Let us denote the algebra of complex split quaternion matrices byH^C_S:H^C_S can be defined with the algebra of22complex matrices:

H^C_SD

P0E0CP1E1CP2E2CP3E3D

P0CiP1 P2CiP3 P2 iP3 P0 iP1

WP0; P1; P2; P32C

(3.7) For any P DP0E0CP1E1CP2E2CP3E32H^C_S, we define S_P DP0E0, the scalar matrix part ofP; ImP DP1E1CP2E2CP3E3, the imaginary matrix part of P. The conjugate, the complex conjugate and the total conjugate of a complex split quaternion matrix are denoted byP ; P^C, P^C

respectively these are

P DP0E0 P1E1 P2E2 P3E3; (3.8) P^C DP0E0CP1E1CP2E2CP3E3 (3.9)

DP0E0 P1E1CP2E2 P3E3; P^C

D.P^C/DP0E0 P1E1 P2E2 P3E3 (3.10)

DP0E0CP1E1 P2E2CP3E3:

Moreover, for anyP DP0E0CP1E1CP2E2CP3E32H^C_S, we can define transpose and adjoint matrix ofP byP^t andAdjP and these are

P^t DP0E0CP1E1CP2E2 P3E3; (3.11) AdjP DP0E0 P1E1 P2E2 P3E3; (3.12)

AdjP DP : (3.13)

The norm of a complex split quaternion matrix P DP0E0CP1E1CP2E2CP3E3D

p11 p12

p₁₂ p₁₁

(3.14) is defined as

kPk D rˇ

ˇˇjp11j² jp12j² ˇ ˇ

ˇ (3.15)

wherep11DP0CiP1andp12DP2CiP3.

Definition 3. A determinant ofP 2H^C_S is defined as

detP DP₀²detE0CP₁²detE1CP₂²detE2CP₃²detE3: (3.16) From the determinant of a complex split quaternion basis can be written as

detP DP₀²CP₁² P₂² P₃²: (3.17) Theorem 4. For anyP; Q2H^C_S and 2Cthe following properties are satisfied:

.i /detP Ddet.P /Ddet.P^C/Ddet P^t

; .i i /det. P /D ²detP;

.i i i /det.PQ/DdetPdetQ:

Proof. .i /ForP DP0E0CP1E1CP2E2CP3E32H^C_S, from (3.17) we get detP Ddet.P /Ddet.P^C/Ddet P^t

DP₀²CP₁² P₂² P₃²: .i i / For any 2CandP DP0E0CP1E1CP2E2CP3E32H^C_S;

P D. P0/ E0C. P1/ E1C. P2/ E2C. P3/ E3: Thus,

det. P /D ²P₀²C ²P₁^{2 2}P₂^{2 2}P₃² D ² P₀²CP₁² P₂² P₃² D ²detP:

.i i i /LetP DP0E0CP1E1CP2E2CP3E3and

QDQ0E0CQ1E1CQ2E2CQ3E3be complex split quaternion matrices, then PQ

D.P0Q0 P1Q1CP2Q2CP3Q3/ E0C.P0Q1CP1Q0 P2Q3CP3Q2/ E1

C.P0Q2CP2Q0 P1Q3CP3Q1/ E2C.P0Q3CP3Q0CP1Q2 P2Q1/ E3

and from (3.17) the determinant ofPQcan be found as in the form of det.PQ/

DP₀²Q²₀CP₀²Q²₁CP₁²Q²₀CP₁²Q²₁CP₂²Q²₂CP₂²Q₃²CP₃²Q₂²CP₃²Q₃² P₀²Q²₂ P₀²Q²₃ P₁²Q²₂ P₁²Q₃² P₂²Q²₀ P₂²Q²₁ P₃²Q²₀ P₃²Q₁²: On the other hand, the determinants ofP andQareP₀²CP₁² P₂² P₃²andQ²₀C Q₁² Q²₂ Q²₃respectively

detPdetQ

DP₀²Q²₀CP₀²Q²₁CP₁²Q²₀CP₁²Q²₁CP₂²Q²₂CP₂²Q₃²CP₃²Q₂²CP₃²Q₃² P₀²Q²₂ P₀²Q²₃ P₁²Q²₂ P₁²Q₃² P₂²Q²₀ P₂²Q²₁ P₃²Q²₀ P₃²Q₁²: Thus,

det.PQ/DdetPdetQ:

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 mat- rix.

P^C P^C

D P^C

P^C D.P₀²CP₁² P₂² P₃²/E0 (3.18) and from (3.17) the determinant ofP^C PC

is found as det.P^C P^C

/D.P₀²CP₁² P₂² P₃²/²: (3.19) Hence, the determinant of a complex split quaternion matrix can be written as

.detP /²Ddet.P^C PC

/: (3.20)

If detP ¤0;the inverse of a complex split quaternion matrix is defined as P ¹D 1

detPP : (3.21)

From (3.8), (3.17) and (3.21) can be written

Q ¹D 1

P₀²CP₁² P₂² P₃².P0E0 P1E1 P2E2 P3E3/ : (3.22)

Example4. LetP D

1 i 2

E0C

1 i 2

E1 1

2E2C₂ⁱE3be a complex split qua- ternion matrix. Then, P can be written as P D

1 1 0 i

and from(3.22), the inverse ofP is calculated as

P ¹D 1Ci

2

E0C

1 i 2

E1Ci

2E2C1 2E3

D 1 i

0 i

:

Theorem 5. Complex split quaternion matrices satisfy the following properties forP 2H^C_S.

.i / E1cDcE1,E2cDcE2,E3cDcE3for any complex numberc, .i i / P²DS_P² det.ImP / E0C2SPImP;

.i i i /Every complex split quaternion matrix P can be uniquely expressed as P D Z1CZ2E2;whereZ1,Z22M2.C/:

Theorem 6. For anyP; Q2H^C_S the following properties are satisfied:

.i /detP D kPk²; .i i / P ¹

D P ¹

ifP is invertible, .i i i / P^C 1

D P ¹C

ifP is invertible, .iv/

h

Pti 1

Dh

P ¹it

ifP is invertible, .v/ .PQ/^C DP^CQ^C;

.vi / .PQ/ ¹DQ ¹P ¹ifP andQare invertible.

The proof is analogous to the proof of Theorem3.

Example5. LetP DE0CE2CE3andQDE0CE1.Then, .i / .PQ/^C DE0 E1C2E2¤E0 E1 2E3DQ^CP^C, .i i / P .P /^C DE0 2E1C2E3¤E0C2E1C2E3D.P /^CP.

With these examples we get the following Corollary for complex split quaternion matrices.

Corollary 2. LetP; Q2H^C_S:Then the followings are satisfied:

.i / .PQ/^C ¤Q^CP^C in general;

.i i / P .P /^C ¤.P /^CP in general.

Definition 4. For anyP DP0E0CP1E1CP2E2CP3E32H^C_S,

.i /if off-diagonal entries ofP are0thenP is calleda diagonal matrixandP is in

form ofP0E0CP1E1,

.i i / if P^t DP then P is called a symmetric matrix and P is in form of P D P0E0CP1E1CP2E2,

.i i i / if P^t DP ¹ then P is called an orthogonal matrix and P is in form of P DP0E0CP3E3and detP D1,

.iv/ if P^t

DP then P is called a Hermitian matrix and P is in form of P D P0E0CP3E3,

.v/ if Pt

DP ¹ then P is called an unitary matrix andP is in form of P D P0E0CP1E1CP2E2and detP D1.

4. CONCLUSION

This paper has investigated the main properties of complex quaternion and com- plex 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.

REFERENCES

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[3] M. Erdo˘gdu and M. ¨Ozdemir, “On complex split quaternion matrices.”Adv. Appl. Clifford Algebr., vol. 23, no. 3, pp. 625–638, 2013, doi:10.1007/s00006-013-0399-z.

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Authors’ addresses

Y. Alag¨oz

Yildiz Technical University, Department of Mathematics, 34220 Istanbul, Turkey E-mail address:ygulluk@yildiz.edu.tr

G. ¨Ozyurt

Yildiz Technical University, Department of Mathematics, 34220 Istanbul, Turkey E-mail address:gozdeozyurt1@gmail.com