CONSIMILARITY OF COMMUTATIVE QUATERNION MATRICES

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In the middle of 19th century, Sir William Hamilton defined the set of real ... After Hamilton had discovered the real quaternions, Segre defined the set of com-.
Miskolc Mathematical Notes Vol. 16 (2015), No. 2, pp. 965–977

HU e-ISSN 1787-2413 DOI: 10.18514/MMN.2015.1421

CONSIMILARITY OF COMMUTATIVE QUATERNION MATRICES ¨ ˇ ¨ HIDAYET HUDA KOSAL, MAHMUT AKYIGIT, AND MURAT TOSUN Received 13 November, 2014 Abstract. In this paper, the consimilarity of complex matrices is generalized for commutative quaternion matrices. In this regard, the coneigenvalue and coneigenvector for commutative quaternion matrices are defined. Also, the existence of solution to the some commutative quaternion matrix equations is characterized and solutions of these matrix equations are derived by means of real representations of commutative quaternion matrices. 2010 Mathematics Subject Classification: 15B33; 15A18 Keywords: commutative quaternion, commutative quaternion matrix, consimilarity, coneigenvalue

1. I NTRODUCTION In the middle of 19th century, Sir William Hamilton defined the set of real quaternions which are denoted by [3] K D fq D q0 C q1 i C q2 j C k´I qs 2 R; s D 0; 1; 2; 3g where i 2 D j 2 D k 2 D 1; ij D j i D k; i k D ki D j; j k D kj D i: There are many applications of these quaternions. One of them is also about matrix theory. The study of the real quaternion matrices began in the first half of the 20th century, [13]. So, Baker discussed right eigenvalues of the real quaternion matrices with a topological approach in [1]. On the other hand, Huang and So introduced left eigenvalues of real quaternion matrices [6]. After that Huang discussed the consimilarity of the real quaternion matrices and obtained the Jordan canonical form of the real quaternion matrices under consimilarity [5]. Jiang and Wei studied the real quae B D C by means of real representation of the real ternion matrix equation X AX quaternion matrices, [8]. Also, Jiang and Ling studied the problem of solution of the e XB D C via real representation of a quaternions quaternion matrix equation AX matrix [7]. c 2015 Miskolc University Press

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After Hamilton had discovered the real quaternions, Segre defined the set of commutative quaternions, [11]. Commutative quaternions are decomposable into two complex variables [2]. The set of commutative quaternions is 4-dimensional like set of quaternions. But this set contains zero-divisor and isotropic elements. There are a lot of works associate with commutative quaternions. Catoni et al. gave a brief survey on commutative quaternions [2]. They introduced functions of commutative quaternionic variables and obtained generalized Cauchy-Riemann conditions. Geometrical introduction of commutative quaternions were considered by Severi in association with functions of two complex variable [12]. Differential properties of commutative quaternion functions were studied by Scorza-Drogoni [10]. Also, Kosal and Tosun considered commutative quaternion matrices. Moreover, they investigated commutative quaternion matrices using properties of complex matrices [9]. 2. A LGEBRAIC PROPERTIES OF COMMUTATIVE QUATERNIONS A set of commutative quaternions is denoted by [2] H D fq D t C xi C yj C ´k W t; x; y; ´ 2 R and i; j; k … Rg where i 2 D k 2 D 1; j 2 D 1; ij D j i D k; i k D ki D j; j k D kj D i: There exist three kinds of conjugate of q D t C xi C yj C ´k, qDt

xi C yj

´k; e q D t C xi

yj

´k; e qDt

xi

yj C ´k

and the norm is defined by q :e q kqk4 Dh q: q :e

D .t C y/2 C .x C ´/2

ih

.t

y/2 C .x

i ´/2 :

Multiplication of any commutative quaternions q D t C xi C yj C ´k and q1 D t1 C x1 i C y1 j C ´1 k are defined in the following ways, qq1 D .t t1 xx1 C yy1 ´´1 / C .xt1 C t x1 C ´y1 C y´1 / i : C .t y1 C yt1 x´1 ´x1 / j C .´t1 C t ´1 C xy1 C yx1 / k It is nearby to identify a commutative quaternion q 2 H with a real vector q 2 R4 : Such an identification is denoted by 0

1 t B x C C q D t C xi C yj C ´k Š q D B @ y A: ´

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Then multiplication of q and q1 can be shown by using ordinary matrix multiplication 0 1 0 1 t x y ´ t1 B x t B C ´ y C C : B x1 C qq1 D q1 q Š Lq q1 D B @ y A @ ´ t x y1 A ´ y x t ´1 where Lq is called the fundamental matrix of q. Theorem 1 ([9]). If q and q1 are commutative quaternions and 1 ; 2 are real numbers, then the following identities hold: 1/ q D q1 , Lq D Lq1 2/ LqCq1 D Lq C Lq1 3/ L1 qC2 q1 D 1 Lq C 2 Lq1    1 4/ T race Lq D q C q Ce q Ce q D 4t; kqk D det Lq 4 : Theorem 2 ([9]). Every commutative quaternion can be represented by a 2x2 matrix with complex entries. Proof. Let q D t C xi C yj C ´k 2 H; then every commutative quaternion can be uniquely expressed as q D c1 C c2 j where c1 D t C xi; c2 D y C ´i are complex numbers. The linear map 'q W H ! H is defined by 'q .p/ D qp for all p 2 H: This map is bijective and 'q .1/ D c1 C c2 j 'q .j / D c2 C c1 j with this transformation, the commutative quaternions can be seen as subsets of the matrix ring M2 .C/ ; the set of 2x2 matrices  ND

c1 c2 c2 c1

Then, H and N are essentially same.



 W c1 ; c2 2 C : 

Definition 1. Two commutative quaternions q and q1 are said to be consimilar if there exists a commutative quaternion p; kpk ¤ 0 such that p q p 1 D q1 I this is c written as q  q1 : Theorem 3. For three commutative quaternion q; q1 ; q2 2 H; the following statements hold: c Reflexive: q  q: c c Symmetric: if q  q1 ; then q1  q; c c c Transitive: if q  q1 ; q1  q2 ; then q  q2 :

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Proof. Reflexive 1 q 1 1 D q trivially, for q 2 H: So, consimilarity is reflexive. Symmetric: Let p q p 1 D q1 : As p is nonsingular, we have  1  1 p q1 p D p p q p 1 p D q: So, consimilarity is symmetric. Transitive: Let p1 q p1 1 D q1 and p2 q1 p2 1 D q2 : Then q2 D p2 p1 q p1 1 p2 1 D .p2 p1 / q .p2 p1 /

1

: 

So, consimilarity is transitive. c

Then,  is an equivalence relation on commutative quaternions. Obviously the consimilar commutative quaternions have the same norm. 3. C ONSIMILARITY OF COMMUTATIVE QUATERNION MATRICES The set of all m  n commutative quaternion matrices, which is denoted by Hmn , with ordinary matrix addition and multiplication is a ring with unit. For A D aij mn 2    A D af A D af ; e and e are conjugates Hmn ; the matrices A D aij ij ij mn

of A and AT is transpose matrix of A.

B

mn

mn

B

Theorem 4 ([9]). For any A 2 Hmn and B 2 Hns ; the followings are satisfied:  T    A D i. A D AT ; e

AT

 T ; e A D

AT

;

ii..AB/T D B T AT ; 1 iii.If A;B 2 Hnn are  nonsingular,  then  .AB/ D B e e; AB D e iv. AB D A B ; A B D e AB A B:

e

e

1A 1;

Definition 2. A matrix A 2 Hnn is said to be similar to a matrix B 2 Hnn if there exists a nonsingular matrix P 2 Hnn such that P 1 AP D B: The relation, A is similar to B, is denoted A  B:  is an equivalence relation on Hnn . Definition 3. A matrix A 2 Hnn is said to be consimilar a matrix B 2 Hnn if there exists a nonsingular matrix P 2 Hnn such that P AP 1 D B: The relation, A c c is consimilar to B, is denoted A B:  is an equivalence relation on Hnn . Clearly if A 2 Cnn , then A D A Thus, if A 2 Cnn is consimilar to B 2 Cnn as complex matrices, A is consimilar to B as commutative quaternion matrices. Then, consimilarity relation in Hnn is a natural extension of complex consimilarity in Cnn (for complex consimilarity see reference [4]).

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Definition 4. Let A 2 Hnn ;  2 H: If there exists 0 ¤ x 2 Hn1 such that Ax D x then  is called a coneigenvalues of A and x is called a coneigenvector of A associate with : The set of all coneigenvalues is defined as ˚  D  2 H W Ax D x; for some x ¤ 0 : Recall that if x 2 Hn1 .x ¤ 0/, and  2 H satisfying Ax D x, we call x an eigenvector of A, while  is an eigenvalue of A. Theorem 5. If A 2 Hnn is consimilar to B 2 Hnn then, coneigenvalues of A and B are the same. c

Proof. Let A B: Then, there exists a nonsingular matrix P 2 Hnn such that B D P AP 1 : Let  2 H be a coneigenvalue for the matrix A; then we can find a matrix x 2 Hn1 such that Ax D x; x ¤ 0: Let y D P x: Then By D P AP 1 y D P AP 1 P x D P Ax D P x D y:  Theorem 6. Let A 2 Hnn ; then  is coneigenvalue of A if and only if for any 0 ¤ kˇk ; ˇˇ 1 is a coneigenvalue of A: Proof. From Ax D x ; we get A xˇ

1



  Dx ˇ

1

 ˇ ˇ

1:



Definition 5 ([9]). Let A D A0 C A1 j 2 Hnn where As 2 Cnn ; s D 0; 1: The 2n  2n matrix   A0 A1 A1 A0 is called the complex adjoint matrix of A and denoted by A . It is nearby to identify a commutative quaternion matrix A 2 Hnn with a complex matrix A 2 C2nn : By the Š symbol, we will denote   2nn A0 A D A0 C A1 j Š A D 2C : A1 Then, the multiplication of A 2 Hnn and B 2 Hnn can be represented by an ordinary matrix product A B Š  .A/ B: Theorem 7. Let A; B 2 Hnn ; the followings are satisfied: i.  .In / D I2n ; ii.  .A C B/ D  .A/ C  .B/ ; iii.  .AB/ D  .A/  .B/ ;  iv. If A is nonsingular, then . .A// 1 D  A 1 :

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¨ ˇ ¨ HIDAYET HUDA KOSAL, MAHMUT AKYIGIT, AND MURAT TOSUN

Theorem 8. For every A 2 Hnn ;  .A/ \ C D  . .A// is the set of coneigenvalues of  .A/ : where  . .A// D f 2 C W  .A/ y D y; f or some y ¤ 0g ; Proof. Let A D A0 C A1 j 2 Hnn where As 2 Cnn ; s D 0; 1 and  2 C be a coneigenvalue of A: Therefore there exists 0 ¤ x 2 Hn1 such that Ax D x: This implies .A0 C A1 j / .x0 C x1 j / D .x0 C x1 j / ; .Ax0 C A1 x1 / D x0  and .A0 x1 C A1 x0 / D x1 : Using these equations, we can write      A0 A1 x0 x D 0 : A1 A0 x1 x1 Therefore, the complex coneigenvalue of the commutative quaternion matrix A is equal to the coneigenvalue of the adjoint matrix  .A/ that is  .A/ \ C D  . .A// :  4. R EAL REPRESENTATION OF COMMUTATIVE QUATERNION MATRICES Let A D A0 C A1 i C A2 j C A3 k 2 Hmn where As 2 Rmn ; s D 0; 1; 2; 3: We will define the linear transformation A .X/ D AX : We can write A .1/ D A0 C A1 i C A2 j C A3 k A .i / D A1 A0 i C A3 j A2 k A .j / D A2 C A3 i C A0 j C A1 k A .k/ D A3 A2 i C A1 j A0 k: Then, we obtain 0

A0 B A1 A D B @ A2 A3

A1 A0 A3 A2

A2 A3 A0 A1

1 A3 A2 C C 2 R4m4n : A1 A A0

Here A is called the representation of A corresponding to the linear transformation A .X/ D AX :

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It is nearby to identify a commutative quaternion matrix A 2 Hmn with a real matrix A 2 R4mn : By the Š symbol,we will denote 0 1 A0 B A1 C 4mn C A D A0 C A1 i C A2 j C A3 k Š A D B : @ A2 A 2 R A3 Then, multiplication of A 2 Hmn and B 2 Hnk can be represented by an ordinary matrix product A B Š A B: Theorem 9. For commutative quaternion matrix A, the following identities are satisfied: i. If A 2 Hmn ; then .1 Pm /

1

A .1 Pn / D A ; Qm1 A Qn D A ;

Rm1 A Rn D A ; Sm1 A Sn D A I where 0

Im B 1P D B 0 m @ 0 0 0

0 0 Im 0 0 Im 0 0

1 0 0 C C; 0 A Im

1 0 0 Im 0 B 0 0 0 Im C C; Rm D B @ Im 0 0 0 A 0 Im 0 0

0

0 B Im Qm D B @ 0 0 0

0 B 0 Sm D B @ 0 Im

Im 0 0 0 0 0 0 Im 0 0 0 Im Im 0 0 0

1 0 0 C C; Im A 0 1 Im 0 C C; 0 A 0

(4.1) ii. If A; B 2 Hmn then ACB D A C B ; iii. If A 2 Hmn ; B 2 Hnr ; in that case AB D A .1 Pn /B D A  B .1 Pr /; iv. If A 2 H mm ; then A is nonsingular if and only if A is nonsingular and .A / 1 D .1 Pm /A 1 .1 Pm /; v. If A 2 Hmm ; S e D "1 1 A "1 A e D "2 1 A "2 A 0 1 0 1 Im 0 0 0 Im 0 0 0 B 0 Im B 0 0 C Im 0 0C C ; "2 D B 0 C; where "1 D B @0 @0 0 Im 0 A 0 Im 0 A 0 0 0 Im 0 0 0 Im mm vi. If A 2 HS ;  .A/ \ C D  .A /

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¨ ˇ ¨ HIDAYET HUDA KOSAL, MAHMUT AKYIGIT, AND MURAT TOSUN

where  .A / D f 2 C W A y D y; f or some y ¤ 0g ; is the set of all eigenvalues of A : Proof. By direct calculation, i., iii. and v. can be easily shown. For now we will prove ii., iv. and vi. ii. Let A D A0 C A1 i C A2 j C A3 k; B D B0 C B1 i C B2 j C B3 k 2 Hmn where As ; Bs 2 Rmn ; s D 0; 1; 2; 3: Then, we have 0

A0 C B0 B A1 C B1 ACB D B @ A2 C B2 0 A3 C B3 A0 A1 B A1 A0 DB @ A2 A3 A3 A2 D A C B

1 A3 C B3 A2 B2 C C A1 C B1 A A0 B0 1 B1 B2 B3 B0 B3 B2 C C B3 B0 B1 A B2 B1 B0

A1 C B1 A2 C B2 A0 B0 A3 C B3 A3 C B3 A0 C B0 A2 B2 1 A10C B1 A2 A3 B0 B B1 A3 A2 C CCB A0 A1 A @ B2 A1 A0 B3

iv. Suppose that A 2 Hmm is nonsingular, From AA AA

1

1

D A Pm A

1

1

D I4 , we have

D  I4

and

A 1 Pm A 1 1 Pm D I4m : Then, A is nonsingular and .A / 1 D 1 Pm A 1 1 Pm : vi. Let A D A0 C A1 i C A2 j C A3 k 2 Hmm where As 2 Rmm ; s D 0; 1; 2; 3 and  2 C be a coneigenvalue of A: Therefore, there exists a nonzero column vector x 2 Hm1 so that Ax D x: Then, we can write A x D x: Then complex coneigenvalue of commutative quaternion matrix A is equivalent to the eigenvalue of A that is  .A/ \ C D  .A / :  5. T HE COMMUTATIVE QUATERNION MATRIX EQUATION X

AX B D C

In this part, we take into consideration the commutative quaternion matrix equation X

AX B D C

A 2 Hmm ;

2 Hnn

(5.1) 2 Hmn :

via the real representation, where B and C the real representation matrix equation of the matrix equation (5.1) by Y

A Y B D C :

We define

(5.2)

Proposition 1. The equation (5.1) has a solution if and only if the equation (5.2) has a solution Y D X :

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Theorem 10. Let A 2 Hmm ; B 2 Hnn and C 2 Hmn : Then the equation (5.1) has a solution X 2 Hmn if and only if the equation (5.2) has a solution Y 2 R4m4n , in which case, if Y is a solution to (5.2), then the matrix: 0 1 In  B i In C 1 C .Im i Im j Im kIm / Y Qm1 Y Qn C Rm1 Y Rn Sm1 Y Sn B XD @ j In A 16 kIn (5.3) is a solution to (5.1). Proof. We show that if the real matrix 0 1 Y11 Y12 Y13 Y14 B Y21 Y22 Y23 Y24 C mn C Y DB ; a; b D 1; 2; 3; 4 @ Y31 Y32 Y33 Y34 A ; Yab 2 R Y41 Y42 Y43 Y44 is a solution to (5.2), then the matrix represented in (5.3) is a solution to (5.1). Since Qm1 Y Qn D Y; Rm1 Y Rn D Y; Sm1 Y Sn D Y; we have  Qm1 Y Qn A Qm1 Y Qn B D C Rm1 Y Rn A Rm1 Y Rn B D C Sm1 Y Sn A Sm1 Y Sn B D C :

(5.4)

The last equation shows that if Y is a solution to (5.2), then Qm1 Y Qn ; Rm1 Y Rn and Sm1 Y Sn are also solutions to (5.2). Thus the undermentioned real matrix:  1 Y Qm1 A Qn C Rm1 A Rn Sm1 A Sn 4 is a solution to (5.2). After calculation , we easily obtain Y0 D

Y00 B Y0 1 Y0 DB @ Y20 Y30 0

Y10 Y00 Y30 Y20

Y20 Y30 Y00 Y10

1 Y30 Y20 C C; Y10 A Y00

(5.5)

(5.6)

where Y00 D 14 .Y11 Y20 D 14 .Y13

Y22 C Y33 Y24 C Y31

Y44 / ; Y10 D 14 .Y12 C Y21 C Y34 C Y43 / ; Y42 / ; Y30 D 14 .Y14 C Y23 C Y32 C Y41 / :

(5.7)

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From (5.7), we formulate a matrix as follows: 0

X D Y00 C Y10 i C Y20 j C Y30 k D

1 .Im i Im j Im 4

1 Im B i Im C C kIm / Y 0 B @ j Im A : kIm

Clearly the real representation of the commutative quaternion matrix X is Y 0 : By Proposition (1), X is a solution to equation (5.1).  Example 1. Solve matrix equation  X

1 i i j



 X

1 0 0 0





2i j 1Cj 1Ci Ck i Cj

D



by using its real representation. Real representation of given equation is 0 X

B B B @

1 0 0 1 0 0 0 0

0 0 1 0 0 1 0 0

0 1 1 0 0 0 0 0

1 0 0 0 0 0 0 1

0 B B DB @

0 0 0 0 1 0 0 1

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

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

0 0 0 1 1 0 0 0

2 1 0 1 0 1 1 0

1

0

C B C B C X B A @

0 1 1 0 0 0 1 1

1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0

1 0 0 1 0 1 2 1

1 1 0 0 1 0 0 1

0 1 1 0 2 1 0 1

0 0 1 1 0 1 1 0

1

0 0 0 1 0 0 1 0

1 1 0 0 1 0 0 1

0 1 0 0 1 0 0 0

0 0 1 1 0 1 1 0

1

0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 0

C C C: A

If we solve this equation, we have 0 B B X D B @

0 0 1 0 0 0 0 1

1 0 0 1 1 1 0 0

1 0 0 0 0 1 0 0

0 1 1 0 0 0 1 1

C C C: A

Then, 0

XD

1 4

I2 i I2 j I2

1 I2    B i I2 C i 1 C j C kI2 X B @ j I2 A D k i C j : kI2

0 0 0 0 0 0 0 0

1 C C C A

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6. A PPENDIX A. C OMMUTATIVE QUATERNION MATRIX EQUATION X AXQ B D C Let

0

1 A0 A1 A2 A3 B A1 A0 A3 A2 C C 2 R4m4n ; A D B @ A2 A3 A0 A1 A A3 A2 A1 A0 Then A is called a real representation of A corresponding to the linear transformae : For A 2 Hmn and B 2 Hnr the following equalities are easy tion LA .X/ D AX to confirm. 2

1

 Pm D e; Qm1 A Qn D A ; A 1 Rm A Rn D A ; Sm1 A Sn D A ;   ii. AB D A 2 Pn B D A e 2 Pr ; B where Q; R; S are given by equation (4.1) and 0 1 Im 0 0 0  B 0 Im 0 0 C 2 C: Pm D B @ 0 0 Im 0 A 0 0 0 Im i.

Pm



A

2

(6.1) (6.2)

Now, we investigate the solution of the matrix equation eB D C (6.3) X AX mm nn mn by its real representation, where A 2 H ;B 2H and C 2 H : We first define the real representation matrix equation (6.3) by Y

A Y B D C :

(6.4)

In the same manner, we have Y D X : Theorem 11. Let A 2 Hmm ; B 2 Hnn and C 2 Hmn : Then the equation (6.3) has a solution X 2 Hmn if and only if the equation (6.4) has a solution Y 2 R4m4n , in which case, if Y is a solution to (6.4), then the matrix: X 0 1 .Im i Im j Im kIm / Y C Qm1 Y Qn D 16 is a solution to (6.3).

Rm1 Y Rn

B Sm1 Y Sn B @

(6.5) 1

Im i Im C C j Im A kIm

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Proof. The proof is a routine process as was performed in Theorem 10. 7. A PPENDIX B. C OMMUTATIVE QUATERNION MATRIX EQUATION X AXQ B D C In the same manner, 0

A0 B A1 'A D B @ A2 A3

A1 A0 A3 A2

A2 A3 A0 A1

1 A3 A2 C C 2 R4m4n A1 A A0

is called real representation of A corresponding to the linear transformation LA .X/ D e: AX For A 2 Hmn and B 2 Hnr the following equalities are easy to confirm. i.

3

Pm



1

'A

3

 Pm D e ; Qm1 'A Qn D 'A ; A

Rm1 'A Rn D 'A ; Sm1 'A Sn D 'A ;   ii. 'AB D 'A 3 Pn 'B D 'A 'e 3 Pr ; B where Q; R; S are given by equation (4.1) and 0 1 Im 0 0 0  B 0 Im 0 0 C 3 C: Pm D B @ 0 0 Im 0 A 0 0 0 Im

Now, we define the real representation matrix equation of the matrix equation eB D C (7.1) X AX by Y

'A Y 'B D 'C

(7.2)

where Y D 'X : Theorem 12. Let A 2 Hmm ; B 2 Hnn and C 2 Hmn : Then the equation (7.1) has a solution X 2 Hmn if and only if the equation (7.2) has a solution Y 2 R4m4n , in which case, if Y is a solution to (7.2), then the matrix: X 0 1 .Im i Im j Im kIm / Y D 16

Qm1 Y Qn

B Rm1 Y Rn C Sm1 Y Sn B @

(7.3) 1

Im i Im C C j Im A kIm

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is a solution to (7.1) . Proof. The proof is a routine process as was performed in Theorem 10.



8. ACKNOWLEDGEMENTS The authors would like to thank the anonymous referees for their helpful suggestions and comments which improved significantly the presentation of the paper. R EFERENCES [1] A. Baker, “Right eigenvalues for quaternionic matrices: a topological approach,” Linear Algebra Appl., vol. 286, pp. 303–309, 1999, doi: 10.1016/S0024-3795(98)10181-7. [2] F. Catoni, R. Cannata, and P. Zampetti, “An introduction to commutative quaternions,” Adv. Appl. Clifford Algebras, vol. 16, pp. 1–28, 2006, doi: 10.1007/s00006-006-0002-y. [3] W. R. Hamilton, Lectures on quaternions. Dublin: Hodges and Smith, 1853. [4] R. A. Horn and C. R. Johnson, Matrix Analysis. UK: Cambridge University Press, 1985. [5] L. Huang, “Consimilarity of quaternion matrices and complex matrices,” Linear Algebra Appl., vol. 331, pp. 21–30, 2001, doi: 10.1016/S0024-3795(01)00266-X. [6] L. Huang and W. So, “On left eigenvalues of a quaternionic matrix,” Linear Algebra Appl., vol. 323, pp. 105–116, 2001, doi: 10.1016/S0024-3795(00)00246-9. [7] T. S. Jiang and S. Ling, “On a solution of the quaternion matrix equation ae x xb D c and its applications,” Adv. Appl. Clifford Algebras, vol. 23, pp. 689–699, 2013, doi: 10.1007/s00006013-0384-6. [8] T. S. Jiang and M. S. Wei, “On a solution of the quaternion matrix equation x ae xb D c and its application,” Acta Math. Sin., vol. 21, pp. 483–490, 2005, doi: 10.1007/s10114-004-0428-x. [9] H. H. K¨osal and M. Tosun, “Commutative quaternion matrices,” Adv. Appl. Clifford Algebras, vol. 24, pp. 769–779, 2014, doi: 10.1007/s00006-014-0449-1. [10] G. Scorza-Dragoni, “The analytic functions of a bicomplex variable (in italian),” Memorie Accademia d’Italia V, vol. 5, p. 597, 1934. [11] C. Segre, “The real representations of complex elements and extension to bicomplex systems,” Math. Ann.), vol. 40, p. 413, 1892, doi: 10.1007/BF01443559. [12] F. Severi, “Opere matematiche,” Acc. Naz. Lincei, Roma, vol. 3, pp. 353–461, 1977. [13] L. A. Wolf, “Similarity of matrices in which the elements are real quaternions,” Bull. Amer. Math. Soc., vol. 42, pp. 737–743, 1936, doi: 10.1090/S0002-9904-1936-06417-7.

Authors’ addresses ¨ K¨osal Hidayet Huda Sakarya University, Faculty of Arts and Sciences, Department of Mathematics, Sakarya, Turkey E-mail address: [email protected] Mahmut Akyiˇgit Sakarya University, Faculty of Arts and Sciences, Department of Mathematics, Sakarya, Turkey E-mail address: [email protected] Murat Tosun Sakarya University, Faculty of Arts and Sciences, Department of Mathematics, Sakarya, Turkey E-mail address: [email protected]