Advances in Pure Mathematics, 2011, 1, 9 15 doi:10.4236/apm.2011.12004 Published Online March 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM On Bicomplex Representation Methods and Applications of Matrices over Quaternionic Division Algebra* Junliang Wu, Pingping Zhang College of Mathematics & Statistics, Chongqing University, Chongqing, Chi na Email: jlwu678@tom.com, zhpp04010248@163.com Received January 6, 2011; revised January 20, 2011; accepted January 25, 2011 Abstract In this paper, a series of bicomplex representation methods of quaternion division algebra is introduced. We present a new multiplication concept of quaternion matrices, a new determinant concept, a new inverse con cept of quaternion matrix and a new similar matrix concept. Under the new concept system, many quaternion algebra problems can be transformed into complex algebra problems to express and study. These concepts can perfect the theory of [J.L. Wu, A new representation theory and some methods on quaternion division algebra, JP Journal of Algebra, 2009, 14(2): 121140] and unify the complex algebra and quaternion division algebra. Keywords: Quaternion Determinant, Product of Quaternion Matrix, Inverse of Quaternion Matrix, Similar Quaternion Matrix, Application, Solution 1. Introduction In recent years, the algebra problems over quaternion division algebra have drawn the attention of mathematics and physics researchers [112]. Quaternion algebra the ory is getting more and more important. In many fields of applied science, such as physics, figure and pattern recognition, spacecraft attitude control, 3D animation, people start to make use of quaternion algebra theory to solve some actual problems. Therefore, it encourages people to do further research [1317] on quaternion alge bra theory and its applications. The main obstacle in the study of quaternion algebra is the noncommutative multiplication of quaternion. Many important conclusions over real and complex fields are different from ones over quaternion division algebra, such as determinant, the trace of matrix multiplication and solutions of quaternion equation. From the conclu sions on quaternion division algebra, we find it to lack for general concepts, such as the definition of quaternion matrix determinant. There are different definitions which are given in [1,3,4,6,11,18] since Dieudonne firstly in troduced the quaternion determinant in 1943. In addition, the inverse of quaternion matrix has not been well de fined so far, because it depends on other algebra con cepts. In the study of quaternion division algebra, people always expect to get some relations between quaternion division algebra and real algebra or complex algebra. However, some conclusions on real or complex fields are correct but not on quaternion division algebra. It makes us to consider establishing other algebra concept system over quaternion division algebra to unify the complex algebra and quaternion division algebra. Recently, Wu in [19] used real representation methods to express quaternion matrices and established some new concepts over quaternion division algebra. From these definitions, we can see that they can convert quaternion division algebra problems into real algebra problems to reduce the complexity and abstraction which exist in all kinds of definitions given in [1,3,6,10,11,20]. However, as Wu in [19] mentioned, these concept system is not suitable for complex algebra. In this paper, based on the bicomplex form of quater nion matrix, we present some new concepts to quaternion division algebra. These new concepts can perfect the theory of Wu in [19] and unify the complex algebra and quaternion division algebra. This paper is organized as follows. In Section 2, we introduce a complex representation method of quaternion matrices and explore the relation between quaternion matrices and complex matrices. In Section 3, we present *This work was supported by National Natural Science Foundation o China (No. 70872123) and Science Research Foundation of Chongqing city of China (0903029).
J. L. WU ET AL. Copyright © 2011 SciRes. APM 10 a series of new concepts over quaternion division algebra and study their properties. In section 4, we establish some important theorems to illustrate the applications and effectiveness of the new concept system. Let C denote the complex field, denote the quaternion set, mn C denote the set of mn complex matrices, mn denote the set of mn quaternion matrices and T denote the transpose matrix of . 2. The Bicomplex Representation Methods of Quaternion Matrices and the Relation between Quaternion Matrices and Complex Matrices For any quaternion matrix mn H, can be uniquely represented as 01 jAA A, (2.1) where 0, 1 mn ss AC , 1jA means to multiply each entries of 1 by from right hand side. For above reasons, we can establish a mapping rela tion between quaternion matrices and complex matrices as follows:s 01 :, mn f AH AA, (2.2) where 0, 1 mn ss AC . The set of mn quaternion matrices is written as and the set of image of is written as img A. Theorem 2.1. Let 01 :, mn f AH AA, 0, 1 mn ss AC . Then the mapping is a bijec tive mapping from to img A. Proof. For any entry 01 , A in img A, there exists the corresponding quaternion matrix 01 jAA A in , therefore is a surjection from to img A. Si multaneously, since any quaternion matrix in can be uniquely represented as the form (2.1), so is an in jection from to img A. Thus is a bijective map ping from to img A. The proof is complete. Theorem 2.2. Bijection 01 :,f AA, 0,1 mn s AC s is an isomorphism mapping from to img A. By the concept of isomorphism mapping, this theorem is easy to prove and we omit it here. □ We shall mention that Theorem 2.2 is the foundation of this article, because isomorphism vector spaces have the same properties. 3. The Bicomplex Matrix Concept System over Quaternion Division Algebra According to the complex representation of quaternion matrices above, a series of new definitions of quaternion division algebra which are helpful to discuss the algebra problems on quaternion division algebra can be given as follows. Definition 3.1. The matrix j EE is said to be a nn unit quaternion matrix if E is a nn unit matrix over complex field. In particular, if 1n , then jj 11 EEE is said to be a unit quaternion writ ten as u a. Definition 3.2. Let and 01 nt j BB BH be given. The operator 00 11 j ABABAB (where 00 11 ,AB AB are both the multiplications of complex matrices) is called the *product of quaternion matrices A and B. In particular, if 1mnt , then we can derive the *product of quaternions. Note: when , mn nt ACBC , then BAB. Under the Definition 3.1 and Definition 3.2, we give some relative properties. For any matrix ,nn AB H, we have: 1) AAEA, where is a nn unit quaternion matrix; 2) BBA; 3) BCACBC; 4) TTT ABB A; 5) Tr Tr BBA. Similarly, we establish a new definition as follows. Definition 3.3. Let 1n H and a be given. Then 00 11 aaaaj XXX is called the *product of quaternion and quaternion vector, where 01 j XX, 11 01 , nn CCXX , 01 aa aj , 01 ,aCaC. Now, we introduce the following concept to quater nion division algebra. Definition 3.4. For any quaternion matrix nn 01 j AA, 01 j AA is said to be the de terminant of , where . is the determinant of a com plex matrix. Note: when nn AC , then A. The Definition 3.3 is reasonable. First of all, the result of a quaternion matrix determinant under Definition3.4 is a quaternion. Secondly, from Definition 3.4 we can see that it can convert the determinant of a quaternion matrix into that of complex matrices to reduce the complexity and abstraction. Finally, the new determinant has the same fundamental properties as that over complex field. That is, if is a nn quaternion matrix andij , then we have 1) T AA. 2) If quaternion matrix B is obtained from quater nion matrix by interchanging two rows (or columns) of , then BA.
J. L. WU ET AL. Copyright © 2011 SciRes. APM 11 3) If quaternion matrix has a zero row (or column), then 0 T AA . 4) n kk AA, wheren n kkk k , k . 5) If the jth row (column) of quaternion matrix equal a multiple of the ith row (column) of the matrix, then 0A. 6) Suppose that , B and C are all nn qua ternion matrices. If all rows of B and C both equal the corresponding to rows (columns) of except that the ith row (column) of equal the sum of the ith of B and C, then BC. 7) If quaternion matrix B is the nn matrix re sulting from adding a multiple of the ith row (or column) of matrix to the jth row (or column) of matrix , then BA. 8) Let and B be nn quaternion matrices re spectively. We have BAB. Up to now, people still treat the inverse matrix concept of quaternion matrix as complex matrix, that is, if qua ternion matrix satisfies 1 AE (where is a real unit matrix), then people think that quaternion ma trix exists its inverse matrix 1 . However, people pointedly ignore two questions. An issue is how to define the product of quaternion matrices 1 and . The other one is how to make a calculation of 1 . It indicates that the terminology of inverse matrix does not have a clear definition in quaternion algebra theory. In the following, we shall give a new definition and specific computational method for the inverse of quater nion matrix. Definition 3.5. Let 01 nn j AA AH be given (where 01 , AA both are complex matrices). If the inverse matrices of 0 A and 1 1 both exist, then quaternion matrix is said to be invertible and the inverse matrix is written as 11 01 j AAA, where 1 0 , 1 1 denote the inverse of complex matrices 0 A, 1 respectively. Note: when nn AC , then 1 A. The inverse of quaternion matrix under the new defi nition has the same fundamental properties as those un der the traditional algebra system. It is easy to show the following facts by the new concept, namely, if a quater nion matrix is invertible, then we have: 1) AA. 2) 11 01 kkk kj AAAA, where k AA A is product of kA which is defined in Definition 3.2. 3) If 12 ,,, m AA A are all invertible quaternion ma trices, then 121 1mmm AAAA AA. Obviously, by the new definition of inverse of quater nion matrix above, people can determine easily whether the inverse matrix of quaternion matrix exists or not and calculate the inverse matrix if possible. Under the definition of inverse of quaternion matrix above, a new concept of similar quaternion matrices can be given as follows: Definition 3.6. Let ,nn AB H, if there exists an invertible quaternion matrix P such that PBP, then and B are said to be similar quaternion ma trices written as B. Note: when ,nn AB C, PBP is equiva lent to 1 00 APBP, where 01 PPP , 01 ,nn PC. For similar quaternion matrices, we will deduce many important properties in the next section. 4. Some Applications of the Bicomplex Matrix Concept System In this section, we establish some important theorems to illustrate the applications and effectiveness of the new concept system for the research of quaternion division algebra. The eigenvalue is an important issue in quater nion division algebra theory, so under the new concept system, we will study firstly the eigenvalues of quater nion matrix and the relation between eigenvalues of similar quaternion matrices in detail. Before showing the application, we’ll introduce firstly some concepts associated with eigenvalue. Definition 4.1. For any matrix nn ij a HA , if there exists nonzero quaternion vector 1n H and a quaternion 01 j (where 0 , 1 are both com plex numbers) such that A X, then is said to be the left eigenvalue of , and is the left ei genvector corresponding to . For the sake of distinction, we call the left eigenvalue and the left eigenvector under Definition 4.1 the left quaternion eigenvalue and the left quaternion eigenvec tor respectively. According to the new definition of quaternion matrix multiplication and A X, we can derive that 0 EAX . Thus f A is said to be the characteristic polynomial of (where the op erator denotes the determinant of quaternion matrix under Definition 3.4). Theorem 4.1. A nn quaternion matrix 01 j AA A (where0 A, 1 A both are complex matri ces), if and are the left eigenvalues of 0 A and 1 A respectively, then aj andbj (aC , bC ) are the left quaternion eigenvalues of . Proof. Since and are the left eigenvalues of 0 A and 1 A respectively, then there exist nonzero vectors
J. L. WU ET AL. Copyright © 2011 SciRes. APM 12 1n C and 1n C such that 01 , AA . We have 01 0 jj aj 0AAA A , for aC . 01 1 jjjjj bj j 0AAA A , for bC . So aj and bj are all the left quaternion ei genvalues of . The proof is complete. □ Similarly, we introduce a new right quaternion eigen value concept. Definition 4.2. For any matrix nn ij a HA , if there exists nonzero quaternion vector 1n and qua ternion 01 j (where01 , are both complex numbers) such that YA Y, then is said to be the right quater nion eigenvalue of , and Y is the right quaternion eigenvector corresponding to . For the right eigenvalue of quaternion matrix, we have the following theorem: Theorem 4.2. A nnquaternion matrix 01 jAA A (where 0 A, 1 Aare both complex matrices), if and are the right eigenvalues of 0 A an 1 A respectively, then aj and bj (,aCbC ) are the right quaternion eigenvalues of . Proof. Since and are the right eigenvalues of 0 A and 1 A respectively, then there exist nonzero vectors 1n C and 1n C such that 01 , AA . We have 01 0 , for jj aja C 0 AAAA . 01 1 ,for jjjjj bj jbC 0 AAAA . So aj and bj are the right quaternion ei genvalues of . The proof is complete. □ Theorem 4.3. If the left eigenvalues of 0 A are 12 ,,, k and the left eigenvalues of 1 A are 12 ,,, m (where 0 A,1 A both are complex matri ces), then the left quaternion eigenvalues of matrix 01 jAA A are saj or , , , 1,,, 1,, t bjaCbCs ktm . Proof. Suppose that is arbitrary left quaternion eigenvalue of , then 0 , 1 01 n j , such that A , that is, 00 00 11 11 A A . Since 0 , we know that both 0 and 1 are not zeroes. So there are two cases as follows: 1) When 0 , obviously, we have 012 ,,, k . So, ,1,2,, iaj ik . 2) When 0 , obviously, we have 112 ,,, m . So, ,1,2,, t bjtm . To sum up 1), 2) and Theorem 4.1, we can draw the conclusion. The proof is complete. □ Theorem 4.4. If the right eigenvalues of 0 A are 12 ,,, k and the right eigenvalues of 1 A are 12 ,,, m (where 0 A,1 A both are complex matri ces), then the right quaternion eigenvalues of matrix 01 j AA A are saj or , , , 1,,, 1,, t baCbCsktm . This proof is similar toTheorem 4.3. So we omit it here. Theorem 4.5. Let nn , then and T have the same quaternion left (right) eigenvalues. Proof. Since 01 j AA A (where 0 nn C,1 nn C A), then 01 TTT j AA . We know i A and T i have the same left (right) eigenvalues (1, 2i ). By Theorem 4.3 and Theorem 4.4, we can draw the conclusion. The proof is complete. □ Theorem 4.6. Let nn and , be given. If is the left (right) quaternion eigenvalue of , then is the right (left) quaternion eigen value of . Proof. Since is the left quaternion eigenvalue of , then there exits nonzero vector such that A . Then TT A , we can have TT T A . So is the right quaternion eigen value of T , by Theorem 4.5, we know is the right quaternion eigenvalue of . The same proof to . So, the proof is complete. □ Specially, when nn AC , if is the left (right) eigenvalue of , then is the right (left) eigen value of . Note: By the new definition of quaternion multiplica tion, the left quaternion eigenvalue of a quaternion ma trix is equivalent to its right quaternion eigenvalue. So they are both called quaternion eigenvalue of the quater nion matrix. In the following, we show an important result. Theorem 4.7. Let ,nn AB H be given. If ∽B, then and B have the same eigenvalues. Proof. Since ∽B, there exists an invertible matrix nn PH such that PBP, that is equivalent to 1 00 APBP and 1 1111 BPP (where 01 jAA A, 01 j BB, 01 j PPP). We know and A 0,1s have the same eigenvalues. By Theorem 4.3 and Theorem 4.4, we can draw that and B have
J. L. WU ET AL. Copyright © 2011 SciRes. APM 13 the same eigenvalues. The proof is complete. □ Theorem 4.8 (The generalized CayleyHamilton theo rem over quaternion division algebra). A quaternion ma trix must be the root of its characteristic polynomial f A. Proof. According to Definition 3.4, we know that: 01 01 01 00110 1 ff j jj jgh j EA EE AA EA EA , where 00 0 g A, 11 1 h A. According to the CayleyHamilton theorem on com plex field, we know 0 g0A, 1 h0A. So, 01 fgh0AAA . It indicates that quaternion matrix must be the root of its characteristic polyno mial f . So, the proof is complete. □ Theorem 4.9. Let (where 01 nn j AA AH 01 ,nn AC) be given. is a diagonalizable matrix if and only if both 0 A and 1 A are diagonalizable ma trices. Proof. is diagonalizable matrix , that is ,there ex its an invertible quaternion matrix P such that PP. It is equivalent to 1 0000 AP P and 1 111 AP P (where 01 j is diagonal matrix). So, is diagonalizable matrix if and only if both 0 A and 1 A are diagonalizable matrices. The proof is complete. □ Corollary 4.9. Let 01 nn j AA AH (where 01 ,nn AC) be given. If 0 A and 1 A both have n different eigenvalues, then is diagonalizable matrix. Corollary 4.9'. Let 01 nn j AA AH (where 01 ,nn AC) be given. Quaternion matrix is di agonalizable matrix if and only if and 1 A both have n linearly independent eigenvactors. Corollary 4.9''. Let 01 nn j AA AH (where 01 ,nn AC) be given. Quaternion matrix is di agonalizable matrix if and only if the geometric multi plicity of 0 A and 1 A both equal their algebraic multi plicity respectively. In Section 3, we have given the new definition of the inverse of quaternion matrix, but that of quaternion is not defined. In fact, a quaternion can be treated as a 11 matrix. So we can define the inverse of quaternion as follows: Definition 4.3. For any quaternion 01 aa aj, if neither of 0 a and 1 a are zeroes, then 11 01 aaaj is said to be the inverse of a, where 10,1 s as is the reciprocal of a. It is easy to verify the following facts. For any ,ab H , we have: 1) uu aaaa a ; 2) abba ; 3) ab cacbc ; 4) 01 nn n aa aj ; 5) If 01 aa aj has the inverse a−, then u aa a . In addition, we discover that there are some special phenomena about the roots of quaternion polynomial under the new definition of quaternion multiplication. Definition 4.4. The polynomial which has the form as follows: *1 *0*1 *0 01 1 nnn axaxax ax is said to be quaternion polynomial with complex coeffi cients (where , 0,1,, i ai n are all complex numbers, 01 xxj , *0 is the *product of iquaternion and 0 is unit quaternion). Theorem 4.10. Let x be a quaternion polyno mial with complex coefficients. Then x has infinite quaternion roots. Proof. By Fundamental Theorem of algebra, x exists at least one complex root 0 , then for any given complex number 1 , obviously, 01 xj is the root of x. The proof is complete. □ Theorem 4.11. Let x be a quaternion polyno mial with complex coefficients and 01 nn j AA AH be a given quaternion matrix (where, both 0 A and 1 A are compex matrices). If is the eigenvalue of 0 A, then f is the eigenvalue of f . Proof. According to the new definition of quaternion multiplication, we can easily obtain 0 ff A. Since is the eigenvalue of 0 A, so f is the ei genvalue of 0 f . The proof is complete. □ Under the new concept system, we can also solve the problems of existence and uniqueness of the solutions to the quaternion system of linear equations A b, where operator ‘ ’ denotes the new multiplication of quaternion matrices. As we known, for any mn , can be repre sented uniquely as 01 j AA A, where A 0,1s are nn complex matrices. Let T 1011202101 ,,, nn xjx xjx xj X and T 1011 202101 ,,, nn bbjbbj bbj b be 1n quater nion vectors, then the following theorems are valid. Theorem 4.12. Let 01 nn j AA AH be given and 01 j XX be 1n quaternion vector. If rank s r A and the fundamental system of solutions to the system of homogeneous linear equations ss 0AX is 1,2 ,, i ii in r 0,1s respectively, then any solu tion to the quaternion system of homogeneous linear
J. L. WU ET AL. Copyright © 2011 SciRes. APM 14 equations 0A can be expressed as follows: 00 11 01 01020200 11 11121211 nrnr nr nr cc c cc cj X , where , 0,,, 0,1 s st ss cCtnrs . Proof. By the new definition of quaternion matrix multiplication, the quaternion system of homogeneous linear equations 0AX is equivalent to the system of homogeneous linear equations 00 11 0 0 X X A A. Since any solution to the system of homogeneous linear equations ss 0AX can expressed as 00 1122sssss sn rsn r cc c X (where , 1,,, 0,1 s st ss cCtnrs ) and the solutions of the quaternion system of homoge neous linear equations 0A are 01 j XX. So we can draw the conclusion. So, the proof is complete. □ Corolla ry 4.12. Let 01 jAA A be a given quater nion matrix (where mn s AC , 0,1s). If rank 0 = rank 1 = n, then the quaternion system of homogeneous linear equations 0A has unique solution T 0, 0,00X. Corollary 4.12'. Let 01 jAA A be a given qua ternion matrix (wheremn s AC ,0,1s). If rank 0nA and rank 1nA, then the quaternion sys tem of homogeneous linear equations 0A only exists complex solutions. Theorem 4.13. Let 01 jAA A be a given quater nion matrix, 01 j XX and 01 jbb b be qua ternion vectors (where mn s AC, 1n s XC, T 12 ,,, ssssn bb bb, st bC, 0,1s, 1, 2,,tn). If there is at least one 00,1s such that rank 0 s rank 00 ss b, then the quaternion system of linear equations A b has no solution. Proof. By the new definition of quaternion matrix multiplication, the quaternion system of linear equations A b is equivalent to the system of linear equa tions 00 0 11 1 b b A A X, since 000 rank ra<nk sss Ab, the system of linear equations 00 0 11 1 b b A A X have no solution, that is, the quaternion system of linear equations A b has no solution. So, the proof is complete. □ Theorem 4.14. Let 01 jAA A be a given quater nion matrix and 01 j XX be a given quaternion vector (where mn s AC, 1n s XC, 0,1s). We sup pose that the fundamental system of solutions to the sys tem of linear equations ss 0AX is 1,2 ,, ss nr 0,1s respectively and 0,1 ss is a special so lution of the system of linear equations ss Xb re spectively, and rank = rank s Ab 0, 1s, then any solution to the quaternion system of linear equations A b can be expressed as: 00 11 001010202 00 111111212 11 nr nr nr nr cc c cc cj X . Proof. By the new definition of quaternion matrix multiplication, the quaternion system of linear equations A b is equivalent to the system of linear equa tions 00 0 11 1 b b A A X. Since any solution to the system of linear equations ss A b can expressed as follows: 1122 s ssssss nr snr cc c X (where , 1,2,,, 0,1 st s cCtnrs ), so any solution to the system of quaternion linear equations AX b can be expressed as: 00 11 001010202 00 111111212 11 nr nr nr nr cc c cc cj X . The proof is complete. □ Theorem 4.15. Let 01 j AA A be a given quater nion matrix, 01 j XX and 01 jbb b be quater nion vectors (where mn s AC, 1n s XC, 1n sC b, 0,1s). If rank = rank s Ab = n 0,1s, then the quaternion system of linear equations A b exists unique solution. Proof. By the new definition of quaternion matrix multiplication, the quaternion system of linear equations Xb is equivalent to the system of linear equations 00 0 11 1 b b A A X, and rank = rank s Ab = n, we know the system of linear equations 00 0 11 1 b b A A X have unique solution. So the quaternion system of linear equa tions A b exists unique solution. The proof is complete. □ Corollary 4.15. Let 01 j AA A be a given nn quaternion matrix and 01 j bb b be a given 1n quaternion vector. If rank = rank s Ab = n 0,1s, then the solution of the quaternion system of equations A b is 1 XA b. Corollary 4.15'. Let mn AC and 01 j bb b (where 1 1 , 0,1, n ss 0bCb ) be given. Then the
J. L. WU ET AL. Copyright © 2011 SciRes. APM 15 quaternion system of linear equations A b has no solution. Corollary 4.15''. Let mn AC and 1n bC be given. If rank = rank Ab , then any solution to the quaternion system of linear equations A b can expressed as 1aj AXb, where aC. 5. References [1] L. X. Chen, “Inverse Matrix and Properties of Double Determinant over Quaternion TH Field,” Science in China (Series A), Vol. 34, No. 5, 1991, pp. 2535. [2] L. X. Chen, “Generalization of CayleyHamilton Theo rem over Quaternion Field,” Chinese Science Bulletin, Vol. 17, No. 6, 1991, pp. 12911293. [3] R, M. Hou, X. Q. Zhao and l. T. Wang, “The Double Determinant of Vandermonde’s Type over Quaternion Field,” Applied Mathematics and Mechanics, Vol. 20, No. 9, 1999, pp. 100107. [4] L. P. Huang, “The Determinants of Quateruion Matrices and Their Propoties,” Journal of Mathematics Study, Vol. 2, 1995, pp. 113. [5] J. L. Wu, L. M. Zou, X. P. Chen and S. J. Li, “The Esti mation of Eigenvalues of Sum, Difference, and Tensor Product of Matrices over Quaternion Division Algebra,” Linear Algebra and its Applications, Vol. 428, 2008, pp. 30233033. doi:10.1016/j.laa.2008.02.008 [6] T. S. Li, “Properties of Double Determinant over Quater nion Field,” Journal of Central China Normal University, Vol. 1, 1995, 37. doi:10.1007/BF02652076 [7] B. X. Tu, “Dieudonne Determinants of Matrices over a Division Ring,” Journal of Fudan university, 1990A, Vol. 1, pp. 131138. [8] B. X. Tu, “Weak Direct Products and Weak Circular Product of Matrices over the Real Quaternion Division Ring,” Journal of Fudan University, Vol. 3, 1991, p. 337. [9] J. L. Wu, “Distribution and Estimation for Eigenvalues of Real Quaternion Matrices,” Computers and Mathematics with Applications, Vol. 55, 2008, pp. 19982004. doi:10.1016/j.camwa.2007.07.013 [10] B. J. Xie, “Theorem and Application of Determinants Spread out of SelfConjugated Matrix,” Acta Mathe matica Sinica, Vol. 5, 1980, pp. 678683. [11] Q. C. Zhang, “Properties of Double Determinant over the Quaternion Field and Its Applications,” Acta Mathe matica Sinica, Vol. 38, No. 2, 1995, pp. 253259. [12] W. J. Zhuang, “Inequalities of Eigenvalues and Singular Values for Quaternion Matrices,” Advances in Mathe matics, Vol. 4, 1988, pp. 403406. [13] W. Boehm, “On Cubics: A Survey, Computer Graphics and Image Processing,” Vol. 19, 1982, pp. 201226. doi:10.1016/0146664X(82)900090 [14] G. Farin, “Curves and Surfaces for Computer Aided Geometric Design,” Academic Press, Inc., San Diego CA, 1990. [15] K. Shoemake, “Animating Rotation with Quaternion Calculus,” ACM SIGGRAPH, 1987, Course Notes, Com puter Animation: 3–D Motion, Specification, and Con trol. [16] Q. G. Wang, “Quaternion Transformation and Its Appli cation to the Displacement Analysis of Spatial Mecha nisms, Acta Mathematica Sinica, Vol. 15, No. 1, 1983, pp. 5461. [17] G. S. Zhang, “Commutativity of Composition for Finite Rotation of a Rigid Body,” Acta Mechanica Sinica, Vol. 4, 1982. [18] E. T. Browne, “The Characteristic Roots of a Matrix,” Bulletin of the American Mathematical Society, Vol. 36, 1930, pp. 705710. doi:10.1090/S000299041930050417 [19] J. L. Wu and Y. Wang, “A New Representation Theory and Some Methods on Quaternion Division Algebra,” Journal of Algebra, Vol. 14, No. 2, 2009, pp. 121140. [20] Q. W. Wang, “The General Solution to a System of Real Quaternion Matrix Equation,” Computer and Mathemat ics with Applications, Vol. 49, 2005, pp. 665675. doi:10.1016/j.camwa.2004.12.002 [21] G. B. Price, “An Introduction to Multicomplex Spaces and Functions,” Marcel Dekker, New York, 1991. [22] D. Rochon, “A Bicomplex Riemann Zeta Function,” To kyo Journal of Mathematics, Vol. 27, No. 2, 2004, pp. 357369. [23] S. P. Goyal and G. Ritu, “The Bicomplex Hurwitz Zeta function,” The South East Asian Journal of Mathematics and Mathematical Sciences, 2006. [24] S. P. Goyal, T. Mathur and G. Ritu, “Bicomplex Gamma and Beta Function,” Journal of Raj. Academy Physical Sciences, Vol. 5, No. 1, 2006, pp. 131142. [25] J. N. Fan, “Determinants and Multiplicative Functionals on Quaternion Matrices,” Linear Algebra and Its Appli cations, Vol. 369, 2003, pp. 193201. doi:10.1016/S00243795(02)00722X [26] Q. W. Wang, “A System of Four Matrix Equations over Von Neumann Regular Rings and It Applications,” Acta Mathematica Sinica, Vol. 21, 2005, pp. 323334. doi:10.1007/s1011400404931 [27] Q. W. Wang, “A System of Matrix Equation and a Linear Matrix Equation over Arbitrary Regular Rings with Iden tity,” Applied Linear Algebra, Vol. 384, 2004, pp. 4354. doi:10.1016/j.laa.2003.12.039 [28] W. J. Zhuang, “The Guide of Matrix Theory over Qua ternion Field,” Science Press, Beijing, 2006, pp. 150. [29] W. L. LI, “Quaternion Matrices,” “National Defense Science and Technology University,” Vol. 6, 2002, pp. 7374. [30] R. X. Jiang, “Linear Algebra,” People’s Educational Press, China, 1979, pp. 4142.
