Real Matrix Representation of Split Quaternion

Real Matrix Representation of Split Quaternion Matrices with Applications Melek Erdo¼ gdu

Mustafa Özdemir

November 13, 2013

Abstract In this paper, we investigate some important properties of split quaternion matrices by using real matrices. Firstly, we introduce real matrix representation of a split quaternion matrix. Then we prove the existence of left eigenvalue of a split quaternion matrix. Moreover, we give the relation between complex left eigenvalues of a split quaternion matrix and eigenvalues of its real matrix representation. Using this relation, we state how to …nd eigenvectors of a split quaternion matrix corresponding to complex left eigenvalues. Furthermore, we state a real linear system which is equaivalent to right eigenvalue equation. Finally, we give a method to solve linear split quaternionic equations of the form Ax = b by using obtained results. Keywords: Split Quaternion, Split Quaternion Matrix, Cli¤ord Algebra. AMS Subject Classi…cations: 15A66, 51B20.

1

Introduction

Quaternion algebra was introduced by Irish Mathematician Sir William Rowan Hamilton in 1843. It was a number system that extends the complex numbers. The set of quaternions can be represented as H = fq = q1 + q2 i + q3 j + q4 k; q1 ; q2 ; q3 ; q4 2 Rg: The imaginary units satisfy i2 = j 2 = k 2 =

1 and ijk =

1:

Quaternions form a four dimensional associative normed division algebra over real numbers. Furthermore, the quaternions were the …rst noncommutative division algebra to be discovered [6], [12], [2]. In 1849, James Cockle introduced the set coquaternions. The real algebra of coquaternions, b is a four dimensional vector space over the real …eld R of real numbers with a basis denoted by H; f1; i; j; kg satisfying i2 = 1; j 2 = k 2 = 1 and ijk = 1: Soon after discovery of algebra of coquaternions, the coquaternions came to be called split quaternions due to the division of units into positive and negative terms. Like quaternions, the set of split quaternions is noncommutative, but unlike quaternions, it contains zero divisors, nilpotent elements and nontrivial idempotents [7], [10], [3]. Furthermore, any split quaternion can be represented by a 2 2 complex matrix [1]. Both quaternions and split quaternions …nd uses in theoretical and applied mathematics. There are some studies related to geometric applications of split quaternions such as [7], [8] and [9]. For example, any timelike unit split quaternion represents a Lorentzian rotation [9], [11]. Corresponding Author

1

Particularly, the geometric and physical applications of quaternions require solving quaternionic equations. Therefore, there are many studies on quaternionic and split quaternionic equations. For example; in [5], the method of solving linear split quaternionic equations involving axb is given and in work [10], De Moivre’s formula is used to …nd the roots of split quaternion as well. The matrices with split quaternion entries are …rst investigated in [1] and [3]. Both studies are used complex matrices. In the study [1], a brief summary of split quaternion and some essential properties of split quaternion matrices are given. Then, a method for …nding complex right eigenvalues and left eigenvalues of the split quaternion matrix is given in [3]. Moreover, Gershgorin theorems are stated for right and left eigenvalues of split quaternion matrices in [3]. Furthermore, the complex split quaternion and their matrices are investigated in [4]. The main purpose of this paper is to examine split quaternion matrices by using real matrices. For this purpose, we …rst present a brief introduction of split quaternions and their matrices. Then, we de…ne real matrix representation of a split quaternion matrix. Moreover, we prove the existence of left eigenvalue of a split quaternion matrix with the relation between complex left eigenvalues of a split quaternion matrix and eigenvalues of its real matrix representation. By this relation, we state the way of …nding eigenvectors of a split quaternion matrix corresponding to complex left eigenvalues. Furthermore, we state a real linear system which is equivalent to right eigenvalue equation. Finally, we give a method to solve linear split quaternionic equations of the form Ax = b by using the obtained results and we solve some linear split quaternionic equations by this way.

2

Preliminaries

In this section, we present an introduction to split quaternions and their matrices for the necessary background [7], [10], [1], [3].

2.1

Split Quaternions The set of split quaternions can be represented as b = fq = q1 + q2 i + q3 j + q4 k; q1 ; q2 ; q3 ; q4 2 Rg H

where the product table is:

1 i j k

1 1 i j k

i i 1 k j

j j k 1 i

k k j : i 1

! We write any split quaternion in the form q = (q1 ; q2 ; q3 ; q4 ) = Sq + Vq where Sq = q1 denotes the ! scalar part of q and Vq = q2 i + q3 j + q4 k denotes vector part of q: If Sq = 0 then q is called pure split quaternion and the set of pure quaternions can be identi…ed with Lorentzian space. Here, the Lorentzian space is Euclidean space with Lorentzian inner product h! u;! v iL =

u1 v1 + u2 v2 + u3 v3

where ! u = (u1 ; u2 ; u3 ); ! v = (v1 ; v2 ; v3 ) 2 E3 and denoted by E31 . And the rotations in Lorentzian space can be stated with split quaternions such as expressing the Euclidean rotations using quaternions, [9]. b is denoted by q and it is The conjugate of a split quaternions q = q1 + q2 i + q3 j + q4 k 2 H q = Sq

! Vq = q 1

q2 i

2

q3 j

q4 k:

The sum and product of split quaternions p and q are de…ned as ! ! p + q = Sp + Sq + V p + V q ; D ! !E ! ! ! pq = Sp Sq + Vp ; Vq + Sp Vq + Sq Vp + Vp L

respectively. Here h , iL and

L

! Vq ;

denote Lorentzian inner and vector product, that is

L

h! u;! v iL = ! u

L

u1 v1 + u2 v2 + u3 v3 ;

! v =

e1 u1 v1

e2 u2 v2

e3 u3 v3

;

for vectors ! u = (u1 ; u2 ; u3 ) and ! v = (v1 ; v2 ; v3 ) of Lorentzian space, respectively. And the norm of a split quaternion q is de…ned by q p Nq = jqqj = jq12 + q22 q32 q42 j: If Nq = 1 then q is called unit split quaternion and q0 = q=Nq is a unit split quaternion for Nq 6= 0: And the product Iq = qq = qq = q12 + q22 q32 q42 determines the character of a split quaternion. if Iq < 0; then q is called spacelike; if Iq > 0; then q is called timelike; if Iq = 0; then q is called lightlike (null). b the following properties are satis…ed Theorem 1. For any p; q; r 2 H, i. qq = qq; _

ii. jc = cj, 8c 2 C; iii. q 2 = Sq2

! Vq

2

! + 2Sq Vq ;

iv. (pq) = q p; v. (pq)r = p(qr); vi. pq 6= qp in general ; vii. q = q if and only if q 2 R; viii. If q12 + q22 6= q32 + q42 then q

1

=

q

2;

kqk

b there exists a unique representation of the form q = c1 + c2 j such that c1 ; c2 2 C: ix. 8q 2 H For further information, see [1], [10] and [9].

3

2.2

Matrix of Split Quaternions

The set of split quaternion matrices was …rst discussed in [1]. The set of m n matrices b with ordinary matrix addition and with split quaternion entries, which is denoted by Mm n (H); multiplication is a ring with unity. The right and left scalar multiplication are de…ned as Aq = (ast q)

and

qA = (qast );

b and q 2 H, b respectively. For A 2 Mm where A = (ast ) 2 Mm n (H) b p; q 2 H; the followings are satis…ed:

b

n (H);

B 2 Mn

b and

r (H)

q(AB) = (qA)B; (Aq)B = A(qB); (pq)A = p(qA):

b is left (right) vector space over H: b b A = (ast ) 2 Mm n (H) b is the conjugate of A; AT = (ats ) 2 For A = (ast ) 2 Mm n (H); T b b Mn m (H) is the transpose of A; A = (A) 2 Mn m (H) is conjugate transpose of A: For square b split quaternion matrices A; B 2 Mn n (H); if AA = A A then A is called normal matrix; if A = A then A is called Hermitian matrix; if AA = I then A is called unitary matrix; if AB = BA = I then A is called invertible matrix. b is noncommutative algebra, if A 2 Mn n (H) b has right (or left ) inverse then A must Although H b if AB = I then BA = I: For any have a equal left (or right) inverse. Hence for A 2 Mn n (H); b square split quaternion matrix A = A1 + A2 j 2 Mn n (H) where A1 ; A2 2 Mn n (C): The 2n 2n matrix 0 1 A1 A2 @ A A2 A1 n (H)

Mm

is called the complex adjoint matrix of A and denoted by by jAjq = j A j

A.

We de…ne the q determinant of A

where j A j is the usual determinant of the complex matrix A : Theorem 2. The complex adjoint matrices of any A; B 2 Mn i.

In

= I2n ;

ii.

A+B

iii.

AB

= =

A

+

B;

A B;

iv. If A is invertible then ( v.

A

6= (

A)

1

A)

=

A

1

;

in general.

Theorem 3. Let A 2 Mn i. A is invertible;

b

n (H);

then the followings are equivalent:

ii. A! x = 0 has unique solution 0; iii. j

Aj

b has the properties;

n (H)

= 6 0 i.e.

A

is an invertible matrix. 4

Theorem 4.

Let A; B 2 Mn

b

n (H);

then

i. A is invertible if and only if jAjq 6= 0; ii. jABjq = jAjq jBjq consequently, if A is an invertible matrix then A

1

1 q

= jAjq ;

iii. jP AQjq = jAjq for some elementary matrices P and Q: b and 2 H. b If holds the equation A! For a square matrix A 2 Mn n (H) x = ! x (A! x =! x ) ! for some nonzero split quaternion column vector x ; then is called the left (right) eigenvalue of A: The set of the left eigenvalues b : A! =f 2H x = ! x ; for some ! x = 6 0g

l (A)

is called left spectrum of A: Similarly, the set of the right eigenvalues r (A)

b : A! =f 2H x =! x ; for some ! x = 6 0g

is called right spectrum of A: In the study [3]; the existence of right eigenvalues of a split quaternion b is proved with the equality matrix A 2 Mn n (H) r (A)

where (

A)

=f 2C:

A

\C = (

A)

! y = ! y ; for some ! y = 6 0g;

is spectrum of the complex adjoint matrix of A. Hence, any n n split quaternion matrix has at b where most 2n distinct complex right eigenvalues. Moreover, for any A = A1 + A2 j 2 Mn n (H); A1 ; A2 2 Mn n (C); the split quaternion = 1 + 2 j where 1 ; 2 2 C; is a left eigenvalues of A if and only if there exists x1 ; x2 2 Cn such that maxfkx1 k1 ; kx2 k1 g > 0 and 2 3 2 3 32 (A1 0 x1 1 In ) (A2 2 In ) 4 5 = 4 5: 54 x2 0 (A2 2 In ) (A1 1 In )

Here, k! y k1 = maxfkyi k : i = 1; 2; :::; ng where ! y = (y1 ; y2 ; :::; yn ) is a complex column vector. This is a statement of 2n 2n homogeneous complex linear system to …nd left eigenvalues of a split quaternion matrix. For details about split quaternion matrices, see [1] and [3].

3

Real Matrix Representation of a Split Quaternion Matrix Let A and B be any n

n split quaternion matrices. Then we may write A = A1 + A2 i + A3 j + A4 k B = B1 + B2 i + B3 j + B4 k

where As ; Bs 2 Mn as

n (R)

for s = 1; 2; 3; 4: The matrix multiplication of A and B can be written

AB = A1 B1 A2 B2 + A3 B3 + A4 B4 + (A2 B1 + A1 B2 + A4 B3 A3 B4 )i + (A3 B1 + A4 B2 + A1 B3 A2 B4 )j + (A4 B1 A3 B2 + A2 B3 + A1 B4 )k

5

in terms of real matrices. Multiplying plication 2 A1 6 A2 6 4 A3 A4

matrices A and B corresponds to the block matrix multiA2 A1 A4 A3

32 B1 A4 6 B2 A3 7 76 A2 5 4 B3 B4 A1

A3 A4 A1 A2

3

7 7: 5

b and algebra of the Using this correspondence, we may de…ne an isomorphism between Mn n (H) matrices 82 9 3 A1 A2 A3 A4 > > > > > > > : ; A4 A3 A2 A1

b where A1 ; A2 ; A3 ; A4 2 Mn De…nition 1. Let A = A1 + A2 i + A3 j + A4 k 2 Mn n (H) The 4n 4n matrix 2 3 A1 A2 A3 A4 6 A2 A1 A4 A3 7 6 7 4 A3 A4 A1 A2 5 A4 A3 A2 A1

n (R):

is called the real matrix representation of the split quaternion matrix A and denoted by RA : b and r 2 R; the followings are satis…ed ; Proposition 1. For any A; B 2 Mn n (H) i. RA+B = RA + RB ; ii. RAB = RA RB ; iii. RrA = rRA ; iv. RIn = RI4n : Proof. These properties can be proved easily by using properties of real matrices. b and RA be the real matrix representation of matrix of A. We De…nition 2. Let A 2 Mn n (H) de…ne the r determinant of A by jAjr = jRA j : Here jRA j is the usual determinant of the real matrix RA : b then Theorem 5. Let A; B 2 Mn n (H);

i. A is invertible if and only if RA is invertible; that is jAjr 6= 0; ii. jABjr = jAjr jBjr consequently, if A is an invertible matrix then A

1

1 r

= jAjr ;

iii. jP AQjr = jAjr for some elementary matrices P and Q: Proof. Let A and B any n n split quaternion matrices. b such that AB = BA = In : So, i. ()) Suppose A is invertible, then there exists B 2 Mn n (H) we have RAB = RA RB = RIn = I4n : This means RA is also invertible and (RA ) 1 = RA 1 ; that is jAjr = jRA j = 6 0: ()) jAjr = jRA j 6= 0; then RA is also invertible. So, there is a real 4n 4n matrix 2 3 D11 D12 D13 D14 6 D21 D22 D23 D24 7 7 D=6 4 D31 D32 D33 D34 5 D41 D42 D43 D44 6

such that RA D = DRA = I4n : Here Dij 2 Mn 2 D11 6 D21 6 4 D31 D41 gives that

n (R)

D12 D22 D32 D42

for i; j = 1; 2; 3; 4: The 32 A1 D13 D14 6 A2 D23 D24 7 76 D33 D34 5 4 A3 A4 D43 D44

equality A2 A1 A4 A3

3 2 A4 In 6 0 A3 7 7=6 A2 5 4 0 A1 0

A3 A4 A1 A2

A1 D11 A2 D21 + A3 D31 + A4 D41 A1 D21 + A2 D11 A3 D41 + A4 D31 A1 D31 + A3 D11 A2 D41 + A4 D21 A1 D41 + A2 D31 A3 D21 + A4 D11

0 In 0 0

0 0 In 0

3 0 0 7 7; 0 5 In

= In ; = 0n ; = 0n ; = 0n :

Using above equations, we see that for B = D11 + D21 i + D31 j + D41 k AB = In : As a consequence of proposition 1, A is also invertible. ii. Using the equality RAB = RA RB ; we have jABjr = jRAB j = jRA RB j = jRA j jRB j = jAjr jBjr : If A is an invertible matrix then 1 = jI4n j = AA

1 r

= jAjr A

1 r

1

1

) A

r

= jAjr :

jP AQjr = jRP AQ j = jRP RA RQ j = jRP j jRA j jRQ j = jRA j = jAjr : b proof of i. of above theorem states the relation between For any invertible A 2 Mn n (H); A 1 and (RA ) 1 ; that is (RA ) 1 = RA 1 : Hence, we may …nd inverse of A by using this relation. Following example explains how to …nd the inverse of a split quaternion matrix by using inverse of its real matrix representation. Example 1. Consider the 2 2 split quaternion matrix

iii.

i+j k 1 i+k

A=

1 k i+j k

:

We may write A = A1 + A2 i + A3 j + A4 k

=

0 1

1 0

+

1 1

So, we …nd the real matrix representation 2 0 1 6 1 0 6 6 1 0 6 6 1 1 RA = 6 6 1 0 6 6 0 1 6 4 1 1 1 1

0 1

i+

1 0

0 1

1 1

j+

1 1

of A as 1 1 0 1 1 1 1 0

0 1 1 0 1 1 0 1 7

1 0 1 1 0 1 1 1

0 1 1 1 1 0 0 1

1 1 1 0 1 1 0 1

1 1 0 1 0 1 1 0

3

7 7 7 7 7 7: 7 7 7 7 5

k:

Here jAjr = jRA j = 9 6= 0. Hence, A is invertible. By using the inverse of RA ; we …nd 2 3 2 3 2 3 2 3 0 1=3 2=3 2=3 0 2=3 1=3 1=3 5+4 5i + 4 5j + 4 5k A 1=4 0 2=3 1=3 2=3 0 2=3 2=3 1=3 2 14 = 3

2i + j i

2k

1 + 2i + 2j 2 + 2i + 2j

Theorem 6. For every A 2 Mn

where

b

k k

3

5:

n (H), l (A)

\ C = (RA );

! ! ! (RA ) = f 2 C : RA X = X ; for some X = 6 0g;

is spectrum of real matrix representation of A. Proof.

Let A = A1 + A2 i + A3 j + A4 k 2 Mn

where A1 ; A2 ; A3 ; A4 2 Mn nonzero column vector

n (R)

and

b

n (H)

2 C be a left eigenvalue of A: Therefore there exists a

! !+x !i + x !j + x !k; x =x 1 2 3 4 ! ! ! ! ! where x1 ; x2 ; x3 ; x4 are real column vectors, such that A x = ! x : This implies !-A x ! ! ! ! ! ! ! A1 x 1 2 2 +A3 x3 +A4 x4 +(A1 x2 +A2 x1 -A3 x4 +A4 x3 )i !-A x ! ! ! ! ! ! ! ! ! ! ! +(A1 x 3 2 4 +A3 x1 +A4 x2 )j+(A1 x4 +A2 x3 -A3 x2 +A4 x1 )k= (x1 +x2 i+x3 j+x4 k): So, we get the following equations ! A x ! ! ! A1 x 1 2 2 + A3 x3 + A4 x4 = !+A x ! A x ! ! A1 x 2 2 1 3 4 + A4 x3 = ! A x ! ! ! A1 x 3 2 4 + A3 x1 + A4 x2 = ! ! ! != A1 x4 + A2 x3 A3 x2 + A4 x 1 Using these equations, we may write 2 A1 A2 6 A2 A1 6 4 A3 A4 A4 A3

A3 A4 A1 A2

32 ! 3 A4 x1 ! 7 6 x A3 7 7 6 !2 7 = A2 5 4 x3 5 ! A1 x 4

!; x 1 !; x 2 ! x3 ; !: x 4 2 ! x1 ! 6 x 6 !2 4 x3 ! x 4

3 7 7 5

Therefore, a complex left eigenvalue of the split quaternion matrix A is equivalent to the eigenvalue of the real matrix representation of A; that is l (A)

\ C = (RA ):

b has at most 4n distinct complex left eigenvalues. Corollary 1. For any A 2 Mn n (H) b corresponding to complex left It is essential to note that the eigenvectors of A 2 Mn n (H) eigenvalues can be found by following the process in proof of above theorem. In the following 8

example, we will …nd the eigenvectors of a split quaternion matrix corresponding to complex left eigenvalues by this way. Example 2. Let i+j j A= j+k i be a split quaternion matrix. In terms of real matrices, we may write A=

0 0

0 0

1 0

+

Then real matrix representation of A 2 0 6 0 6 6 1 6 6 0 RA = 6 6 1 6 6 1 6 4 0 1

0 1

1 1

i+

0 0

j+

0 1

0 0 0 0 0 0 0 1

0 1 1 1 1 0 0 0

0 0 0 0 0 1 0 0

0 0

is found as 0 0 0 1 0 0 0 0

1 0 0 0 0 1 1 1

0 1 0 0 0 0 0 0

1 1 0 1 0 0 1 0

3

7 7 7 7 7 7: 7 7 7 7 5

With long and tedious computations, we get 1;2;3;4

= 0;

k:

5

=

1;

6

= 1;

7

=

i and

8

=i

are eigenvalues of RA : Here, one of eigenvectors of RA corresponding to eigenvalue found as ! T 1 1 1 1 1 0 1 X = 0 :

1;2;3;4

= 0 is

Hence, the real coe¢ cients of the split quaternion eigenvector are 0 1

!= x 1

1 1

!= ; x 2

1 1

!= ; x 3

So, we …nd the one of the eigenvectors of A corresponds to 0 1

! x =

+

1 1

1 1

i+

j+

Similarly, the other eigenvector of A corresponds to ! x =

1 2

1 0

+

1 2

i+

j+

0 1

1;2;3;4

k=

1;2;3;4

1 0

!= ; x 4

0 1

:

= 0 as i+j 1+i j+k

:

= 0 can be found as k=

1

i j+k 2 + 2j

:

b where A1 ; A2 ; A3 ; A4 2 Mn n (R) and Theorem 7. Let A = A1 + A2 i + A3 j + A4 k 2 Mn n (H) !; x !; x !; x ! 2 Rn b = 1 + 2 i+ 3 j + 4 k 2 H be a right eigenvalue of A if and only if there exists x 1 2 3 4 ! ! ! ! such that maxfkx1 k1 ; kx2 k1 ; kx3 k1 ; kx4 k1 g > 0 and 2 32 ! 3 2 ! 3 A1 A2 A3 + 2 In A4 x1 4 In 3 In 1 In 0 6 76 7 7 6 76 ! 7 6 ! 7 6 A2 7 6 x2 7 6 I A I A + I A I 6 7 2 n 1 1 n 4 4 n 3 3 n 0 6 76 7 6 7 6 76 7=6 : 76 ! 7 6 ! 7 6 7 6 A3 7 6 7 I A + I A I A I x 0 6 7 3 n 4 4 n 1 1 n 2 2 n 3 6 76 7 4 5 4 54 5 ! ! 0 A I A I A + I A I x 4

4 n

3

3 n

2

2 n

9

1

1 n

4

T yn is a real column vector. Here, k! y k1 = maxfjyi j : i = 1; 2; :::; ng where ! y = y1 y 2 b Proof. Let A = A1 + A2 i + A3 j + A4 k 2 Mn n (H) where A1 ; A2 ; A3 ; A4 2 Mn n (R) and b be a right eigenvalue of A: Then there exists nonzero split quaternionic = 1 + 2 i+ 3 j + 4 k 2 H !+x !i+ x !j + x !k; where x !; x !; x !; x ! are real column vectors, such that A! vector ! x =x x =! x : 1 2 3 4 1 2 3 4 This is equivalent to the following equations

! A x ! ! ! ! A1 x 1 2 2 + A3 x3 + A4 x4 = x1 !+A x ! A x ! ! ! A1 x 2 2 1 3 4 + A4 x3 = x1 ! A x ! ! ! ! A1 x 3 2 4 + A3 x1 + A4 x2 = x1 ! ! ! !=x ! A1 x4 + A2 x3 A3 x2 + A4 x 1 1 Using these obtained equations, we may write 2 A1 A2 A3 + 4 In 3 In 6 6 6 A2 A1 A4 + 2 In 1 In 6 6 6 6 A3 A4 + 4 In A1 3 In 6 4 A4 A3 A2 + 4 In 3 In

2 In

! +x 2 ! 2 + x2 ! x 3 2 ! + x 4 2

4

A4

! x 3 ! x 1 3 ! 4 + x3 ! x 3 3

3

1 In

4 In

A3

3 In

1 In

A2

2 In

2 In

A1

1 In

! +x 4 ! 4 + x4 ! 1 + x4 ! 2 + x4

2

32 ! x1 76 76 ! 7 6 x2 76 76 76 ! 7 6 x3 76 54 ! x 4

3

1; 3; 2; 1:

2 ! 0 7 6 7 6 ! 7 6 0 7 6 7=6 7 6 ! 7 6 0 7 4 5 ! 0

3

7 7 7 7 7: 7 7 5

!+x !i + x !j + x !k; the real vector De…nition 3. For any split quaternionic vector ! x =x 1 2 3 4 2 ! 3 x1 6 ! 7 7 ! 6 6 x2 7 X =6 ! 7 6 x3 7 4 5 ! x 4 is called the real vector representation of ! x: Theorem 8. For the linear split quaternionic equation ! A! x = b

(1)

! b ! where A 2 Mn n (H); x and b are split quaternionic column vectors, the equation 1 is equivalent to the real equation with 4n unknowns ! ! RA X = B ! ! ! where X and B are the real vector representation of ! x and b ; respectively. !+x !i + x !j + x !k and ! b ! Proof. Let A = A1 + A2 i + A3 j + A4 k 2 Mn n (H); x = x b = 1 2 3 4 ! ! ! ! b1 + b2 i + b3 j + b4 k be split quaternionic column vectors. ! A! x = b ! ! ! ! ! A x ! ! ! , b1 + b2 i + b3 j + b4 k = A1 x 1 2 2 + A3 x3 + A4 x4 !+A x ! A x ! ! ! A x ! ! ! + (A1 x 2 2 1 3 4 + A4 x3 )i + (A1 x3 2 4 + A3 x1 + A4 x2 )j ! ! ! ! + (A x + A x A x + A x )k 1 4

2 3

3 2

4 1

! A x ! ! ! ! A1 x 1 2 2 + A3 x3 + A4 x4 = b1 !+A x ! A x ! ! ! A1 x 2 2 1 3 4 + A4 x3 = b2 , ! A x ! ! ! ! A1 x 3 2 4 + A3 x1 + A4 x2 = b3 !+A x ! A x !+A x !=! A x b 1 4

2 3

3 2

4 1

10

4

2

A1

A2

6 6 6 A2 6 ,6 6 6 A3 6 4 A4

32 ! x1 76 76 ! 6 A3 7 7 6 x2 76 76 ! 6 A2 7 7 6 x3 54 ! x A1 4

A3

A1

A4

A4

A1

A3

A4

A2

3

2 7 7 6 7 6 7 6 7=6 7 6 7 6 7 4 5

3

! b1 ! b2 ! b3 ! b4

7 7 7 ! 7 , RA ! X = B: 7 7 5

! b ! Corollary 2. Let A 2 Mn n (H); x and b be split quaternionic column vectors. ! i. The linear system of equations A! x = b has unique solution if and only if Rank(RA ) = ! Rank(RA ; B ) = 4n: ! ii. The linear system of equations A! x = b has no solution if and only if Rank(RA ) 6= ! Rank(RA ; B ): ! iii. The linear system of equations A! x = b has in…nitely solution if and only if Rank(RA ) = ! Rank(RA ; B ) = t < 4n: In this case the solution depends on 4n t real parameters. ! ! Here (RA ; B ) denotes the augmented matrix of RA with B : Example 3. Consider the following linear system with split quaternionic left coe¢ cients (i + j)x + (1 + i + j + k)y = 1 + 2j ( 1 + j k)x + (i j)y = 2 i

k; k;

where x and y are split quaternionic unknowns. We may write this system as follows; i+j 1+j k Here 2 as

x y

=

1 + 2j 2 i

k k

:

! 2 split quaternion matrix A and the split quaternionic column vector b can be rewritten i+j 1+j k

A=

=

! b = We …nd

1+i+j+k i j

2

6 6 6 6 6 RA = 6 6 6 6 6 4

0 1

1+i+j+k i j

1 0

+

1 0

1 + 2j 2 i

k k

=

0 1 1 0 1 1 0 1

1 0 1 1 1 1 1 0

1 0 0 1 0 1 1 1

1 1

1 1 1 0 1 0 1 1

1 1

i+

1 2 1 1 0 1 0 1 1 0

+

1 1 1 0 1 0 1 1

1 1 0 1 0 1 1 1 1 0 0 1

0 1

j+

i+

1 0 1 1 1 1 1 0

2 0 3

j+

1 0

k;

1 1 2

7 6 7 6 7 6 7 6 7 6 7 and ! 6 B = 7 6 7 6 7 6 7 6 5 4

k:

1 2 0 1 2 0 1 1

! Here Rank(RA ) = 6 and Rank(RA ; B ) = 7: The given system has no solution.

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3

7 7 7 7 7 7: 7 7 7 7 5

Example 4. Consider the following linear system with split quaternionic left coe¢ cients (1 + j k)x + (1 + j)y + (i + j)z = 3 4i 3j (i k)x + (1 + i j)y + (i j 2k)z = 1 + 4i 4j (2 i + 2k)x + (1 j)y + (i 2j + k)z = 3 + 4i 3k;

2k; k;

where x; y and z are split quaternionic unknowns. We may write this system as follows; 2 32 3 2 3 1+j k 1+j i+j x 3 4i 3j 2k 4 i k 1 + i j i j 2k 5 4 y 5 = 4 1 + 4i 4j k 5 : 2 i + 2k 1 j i 2j + k z 3 + 4i 3k Here 3 as

! 3 split quaternion matrix A and the split quaternionic column vector b can be rewritten 2

1+j k i k A=4 2 i + 2k 2

1 =4 0 2

We …nd

2 ! 4 b = 2

6 6 6 6 6 6 6 6 6 RA = 6 6 6 6 6 6 6 6 6 4

1 0 2 0 1 1 1 0 0 1 1 2

1 1 1

1+j 1+i j 1 j

3 2 0 0 0 5+4 1 0 1

3 i+j i j 2k 5 i 2j + k 3 2 0 1 1 1 1 5i + 4 0 0 1 0

1 1 1

3 2 1 1 1 5j + 4 1 2 2

0 0 0

3 2 3 2 3 2 3 2 3 4 3 3 4i 3j 2k 1 + 4i 4j k 5 = 4 1 5 + 4 4 5 i + 4 4 5 j + 4 3 4 0 3 + 4i 3k 1 1 1 0 1 0 1 1 1 0 0 0

0 0 0 1 1 1 1 1 2 0 2 1

0 1 1 1 0 2 1 1 2 1 0 0

0 1 0 1 1 1 0 0 0 1 1 1

1 1 1 0 0 0 0 2 1 1 1 2

1 0 0 1 1 2 1 0 2 0 1 1

1 1 1 0 0 0 1 1 1 0 1 0

1 1 2 0 2 1 0 0 0 1 1 1

1 1 2 1 0 0 0 1 1 1 0 2

0 0 0 1 1 1 0 1 0 1 1 1

0 2 1 1 1 2 1 1 1 0 0 0

3 0 2 5 k; 1 3 2 1 5 k: 3

3

2

7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 ! 6 7; B = 6 6 7 7 6 7 6 7 6 7 6 7 6 7 6 7 6 5 4

3 1 3 4 4 4 3 4 0 2 1 3

3

7 7 7 7 7 7 7 7 7 7: 7 7 7 7 7 7 7 7 5

! ! Here jAjr = jRA j = 42 436 6= 0: A is invertible and the system RA X = B has a unique solution ! X =

0

1

0

0

1

1

1

1

2

1

Hence, the given system has also a unique solution 2 3 2 3 2 3 2 3 2 3 2 x 0 0 1 1 4 y 5 = 4 1 5 + 4 1 5i + 4 1 5j + 4 1 5k = 4 z 0 1 2 2

12

1

T

2

3 k i j + k 5: 2j + 2k

j 1 i

:

References [1] Y. Alagöz, K. H. Oral, S. Yüce, Split Quaternion Matrices. Miskolc Mathematical Notes 13 (2012), 223-232. [2] J. L. Brenner, Matrices of Quaternions. Pasi…c Journal of Mathematics 1 (1951), 329-335. [3] M. Erdo¼ gdu, M. Özdemir, On Eigenvalues of Split Quaternion Matrices. Advances in Applied Cli¤ord Algebras 23 (2013), 615-623. [4] M. Erdo¼ gdu, M. Özdemir, On Complex Split Quaternion Matrices. Advances in Applied Clifford Algebras 23 (2013), 625-638. [5] M. Erdo¼ gdu, M. Özdemir, Two sided Linear Split Quaternionic Equations. Linear and Multilinear Algebra (2013) DOI:10.1080/03081087.2013.851196. [6] I. L. Kantor, A. S. Solodovnikov, Hypercomplex Numbers, An Elementary Introduction to Algebras, Springer-Verlag, 1989. [7] L. Kula, Y. Yayl¬, Split Quaternions and Rotations in Semi Euclidean Space. Journal of Korean Mathematical Society 44 (2007), 1313-1327. [8] M. Özdemir, A.A. Ergin, Some geometric applications of split quaternions. Proc. 16th Int. Conf. Jangjeon Math. Soc. 16 (2005), 108-115. [9] M. Özdemir, A.A. Ergin, Rotations with unit timelike quaternions in Minkowski 3-space, Journal of Geometry and Physics 56 (2006), 322-336. [10] M. Özdemir, The Roots of a Split Quaternion. Applied Mathematics Letters 22 (2009), 258263. [11] M. Özdemir, M. Erdo¼ gdu, H. S ¸im¸sek, On the Eigenvalues and Eigenvectors of a Lorentzian Rotation Matrix by Using Split Quaternions. Advances in Applied Cli¤ord Algebras (2013) DOI 10.1007/s00006-013-0424-2. [12] F. Zhang, Quaternions and Matrices of Quaternions. Linear Algebra and its Applications 251 (1997), 21-57. Melek Erdo¼ gdu Department of Mathematics-Computer Sciences Necmettin Erbakan University 42060 Konya, Turkey e-mail: [email protected]. Mustafa Özdemir Department of Mathematics Akdeniz University 07058 Antalya, Turkey e-mail: [email protected].

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