 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 E-mail: 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): 121-140] 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 [1-12]. 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, 3-D animation, people start to make use of quaternion algebra theory to solve some actual problems. Therefore, it encourages people to do further research [13-17] on quaternion alge-bra theory and its applications. The main obstacle in the study of quaternion algebra is the non-commutative 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  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  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  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 of China (No. 70872123) and Science Research Foundation of Chongqing city of China (09-03-029). 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, H denote the quaternion set, mnC denote the set of mn complex matrices, mnH denote the set of mn quaternion matrices and TA denote the transpose matrix of A. 2. The Bicomplex Representation Methods of Quaternion Matrices and the Relation between Quaternion Matrices and Complex Matrices For any quaternion matrix mnAH, Acan be uniquely represented as 01jAA A, (2.1) where 0, 1mnssAC , 1jA means to multiply each entries of 1A byjfrom right hand side. For above reasons, we can establish a mapping rela-tion between quaternion matrices and complex matrices as follows:s 01:|,mnfAH AA, (2.2) where 0, 1mnssAC . The set of mn quaternion matrices is written as A and the set of image of A is written as imgA. Theorem 2.1. Let 01:|,mnfAH AA, 0, 1mnssAC . Then the mapping f is a bijec- tive mapping from A to imgA. Proof. For any entry 01,AA in imgA, there exists the corresponding quaternion matrix 01jAA A in A, therefore f is a surjection from A to imgA. Si-multaneously, since any quaternion matrix in A can be uniquely represented as the form (2.1), so f is an in-jection from A to imgA. Thus f is a bijective map-ping from A to imgA. The proof is complete. Theorem 2.2. Bijection 01:,fAAA, 0,1mnsAC sis an isomorphism mapping from A to imgA. 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EEE 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 ua. Definition 3.2. Let and 01ntj BB BH be given. The operator 00 11j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 ABAB. Under the Definition 3.1 and Definition 3.2, we give some relative properties. For any matrix ,nnAB H, we have: 1) EAAEA, where E is a nn unit quaternion matrix; 2) ABBA; 3) ABCACBC; 4) TTTABB A; 5) Tr TrABBA. Similarly, we establish a new definition as follows. Definition 3.3. Let 1nXH and aH be given. Then 00 11aaaaj XXXX is called the *-product of quaternion and quaternion vector, where 01jXXX, 1101,nnCCXX, 01aa aj, 01,aCaC. Now, we introduce the following concept to quater-nion division algebra. Definition 3.4. For any quaternion matrix nnAH 01jAAA, 01jAAA is said to be the de-terminant of A, where . is the determinant of a com-plex matrix. Note: when nnAC , then AA. 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 A is a nnquaternion matrix andij, then we have 1) TAA. 2) If quaternion matrix B is obtained from quater-nion matrix A by interchanging two rows (or columns) of A, then BA. J. L. WU ET AL. Copyright © 2011 SciRes. APM 113) If quaternion matrix A has a zero row (or column), then 0TAA . 4) nkkAA, wherennkkk k  , kH. 5) If the jth row (column) of quaternion matrix A equal a multiple of the ith row (column) of the matrix, then 0A. 6) Suppose that A, B and C are all nn qua-ternion matrices. If all rows of B and C both equal the corresponding to rows (columns) of A except that the ith row (column) of A equal the sum of the ith of B and C, then ABC. 7) If quaternion matrix B is the nn matrix re-sulting from adding a multiple of the ith row (or column) of matrix A to the jth row (or column) of matrix A, then BA. 8) Let A and B be nn quaternion matrices re-spectively. We have  ABAB. Up to now, people still treat the inverse matrix concept of quaternion matrix as complex matrix, that is, if qua-ternion matrix A satisfies 1AAE (where E is a real unit matrix), then people think that quaternion ma-trix A exists its inverse matrix 1A. However, people pointedly ignore two questions. An issue is how to define the product of quaternion matrices 1A and A. The other one is how to make a calculation of 1A. 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 01nnj AA AH be given (where 01, AA both are complex matrices). If the inverse matrices of 0A and 11A both exist, then quaternion matrix A is said to be invertible and the inverse matrix is written as 1101j AAA, where 10A, 11A denote the inverse of complex matrices 0A, 1A respectively. Note: when nnAC , then 1AA. 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 A is invertible, then we have: 1) AA. 2) 1101kkkkj AAAA, where kAAA 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 APBP, then A and B are said to be similar quaternion ma-trices written as AB. Note: when ,nnAB C, APBP is equiva-lent to 100APBP, where 01PPPj, 01,nnPPC. 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 nnijaHA , if there exists nonzero quaternion vector 1nXH and a quaternion 01j (where 0, 1 are both com-plex numbers) such that AXX, then  is said to be the left eigenvalue of A, and X 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 AXX, we can derive that 0EAX . Thus fEA is said to be the characteristic polynomial ofA (where the op-erator  denotes the determinant of quaternion matrix under Definition 3.4). Theorem 4.1. A nn quaternion matrix 01jAA A (where0A, 1A both are complex matri-ces), if  and  are the left eigenvalues of 0A and 1A respectively, then aj andbj (aC, bC) are the left quaternion eigenvalues of A. Proof. Since  and  are the left eigenvalues of 0A and 1A 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 0jj aj 0AAA A , for aC . 01 1 jjjjjbj j  0AAA A, for bC . So aj and bj are all the left quaternion ei-genvalues of A. The proof is complete. □ Similarly, we introduce a new right quaternion eigen-value concept. Definition 4.2. For any matrix nnijaHA , if there exists nonzero quaternion vector 1nHand qua-ternion 01j (where01,are both complex numbers) such that YA Y, then  is said to be the right quater-nion eigenvalue of A, 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 nnquaternion matrix 01jAA A (where 0A, 1Aare both complex matrices), if  and  are the right eigenvalues of 0A an 1A respectively, then aj and bj(,aCbC ) are the right quaternion eigenvalues of A. Proof. Since  and  are the right eigenvalues of 0A and 1A respectively, then there exist nonzero vectors 1nC and 1nCsuch that 01,AA . We have 01 0 , for jjaja C  0 AAAA.  01 1 ,forjjjjjbj jbC  0AAAA. So aj and bj are the right quaternion ei-genvalues of A. The proof is complete. □ Theorem 4.3. If the left eigenvalues of 0A are 12,,,k and the left eigenvalues of 1A are 12,,,m (where 0A,1A both are complex matri-ces), then the left quaternion eigenvalues of matrix 01jAA A are sajor , , , 1,,, 1,,tbjaCbCs ktm. Proof. Suppose that  is arbitrary left quaternion eigenvalue of A, then 0, 101nj H, such that A, that is, 00 0011 11AA. 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,,tbjtm . 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 0A are 12,,,k and the right eigenvalues of 1A are 12,,,m (where 0A,1A both are complex matri-ces), then the right quaternion eigenvalues of matrix 01jAA A are saj or , , , 1,,, 1,,tbaCbCsktm. This proof is similar toTheorem 4.3. So we omit it here. Theorem 4.5. Let nnAH, then A and TA have the same quaternion left (right) eigenvalues. Proof. Since 01jAA A (where 0nnAC,1nnCA), then 01TTTjAAA. We know iA and TiA 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AH and ,H be given. If  is the left (right) quaternion eigenvalue of A, then  is the right (left) quaternion eigen-value of A. Proof. Since  is the left quaternion eigenvalue of A, 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 TA, by Theorem 4.5, we know  is the right quaternion eigenvalue of A. The same proof to . So, the proof is complete. □ Specially, when nnAC , if  is the left (right) eigenvalue of A, then  is the right (left) eigen-value of A. 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 A∽B, then A and B have the same eigenvalues. Proof. Since A∽B, there exists an invertible matrix nnPH such that APBP, that is equivalent to 100APBP and 11111ABPP (where 01jAA A, 01jBBB, 01jPPP). We know sB and sA 0,1s have the same eigenvalues. By Theorem 4.3 and Theorem 4.4, we can draw that A and B have J. L. WU ET AL. Copyright © 2011 SciRes. APM 13the same eigenvalues. The proof is complete. □ Theorem 4.8 (The generalized Cayley-Hamilton theo-rem over quaternion division algebra). A quaternion ma-trix A must be the root of its characteristic polynomial fEA. Proof. According to Definition 3.4, we know that: 0101 0100110 1 ff jjjjgh j EAEE AAEA EA, where 00 0gEA, 11 1hEA. According to the Cayley-Hamilton theorem on com-plex field, we know 0g0A, 1h0A. So, 01fgh0AAA. It indicates that quaternion matrix A must be the root of its characteristic polyno-mial f. So, the proof is complete. □ Theorem 4.9. Let (where 01nnj AA AH 01,nnAAC) be given. A is a diagonalizable matrix if and only if both 0A and 1A are diagonalizable ma-trices. Proof. A is diagonalizable matrix , that is ,there ex-its an invertible quaternion matrix P such that APP. It is equivalent to 10000AP P and 1111AP P (where 01j is diagonal matrix). So, A is diagonalizable matrix if and only if both 0A and 1A are diagonalizable matrices. The proof is complete. □ Corollary 4.9. Let 01nnj AA AH (where 01,nnAAC) be given. If 0A and 1A both have n different eigenvalues, then A is diagonalizable matrix. Corollary 4.9'. Let 01nnj AA AH (where 01,nnAAC) be given. Quaternion matrix A is di-agonalizable matrix if and only if and 1A both have n linearly independent eigenvactors. Corollary 4.9''. Let 01nnj AA AH (where 01,nnAAC) be given. Quaternion matrix A is di-agonalizable matrix if and only if the geometric multi-plicity of 0A and 1A 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 01aa aj, if neither of 0a and 1a are zeroes, then 1101aaaj  is said to be the inverse of a, where 10,1sas is the reciprocal of sa. It is easy to verify the following facts. For any ,ab H, we have: 1) uuaaaa a ; 2) abba; 3) ab cacbc; 4) 01nnnaa aj ; 5) If 01aa aj has the inverse a−, then uaa 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 *001 1nnnaxaxax ax  is said to be quaternion polynomial with complex coeffi-cients (where , 0,1,,iai n are all complex numbers, 01xxxj, *0x is the *-product of iquaternion x and 0x is unit quaternion). Theorem 4.10. Let fx be a quaternion polyno-mial with complex coefficients. Then fx has infinite quaternion roots. Proof. By Fundamental Theorem of algebra, fx exists at least one complex root 0x, then for any given complex number 1x, obviously, 01xxj is the root of fx. The proof is complete. □ Theorem 4.11. Let fx be a quaternion polyno-mial with complex coefficients and 01nnj AA AH be a given quaternion matrix (where, both 0A and 1A are compex matrices). If  is the eigenvalue of 0A, then f is the eigenvalue of fA. Proof. According to the new definition of quaternion multiplication, we can easily obtain 0ffAA. Since  is the eigenvalue of 0A, so f is the ei-genvalue of 0fA. 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 AXb, where operator ‘’ denotes the new multiplication of quaternion matrices. As we known, for any mnAH, A can be repre-sented uniquely as 01jAA A, where sA 0,1s are nn complex matrices. Let T1011202101,,,nnxxjx xjx xj X and T1011 202101,,,nnbbjbbj bbj b be 1n quater- nion vectors, then the following theorems are valid. Theorem 4.12. Let 01nnj AA AH be given and 01jXXX be 1n quaternion vector. If rank ssrA and the fundamental system of solutions to the system of homogeneous linear equations ss0AX is 1,2 ,, iii 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 0AX can be expressed as follows: 001101 0102020011 11121211 nrnrnr nrcc ccc cj  X, where , 0,,, 0,1sst sscCtnrs . 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 001100XXAA. Since any solution to the system of homogeneous linear equations ss0AX can expressed as 001122sssss sn rsn rcc c X (where , 1,,, 0,1sst sscCtnrs ) and the solutions of the quaternion system of homoge-neous linear equations 0AX are 01jXXX. So we can draw the conclusion. So, the proof is complete. □ Corolla ry 4.12. Let 01jAA A be a given quater-nion matrix (where mnsAC, 0,1s). If rank 0A = rank1A = n, then the quaternion system of homogeneous linear equations 0AX has unique solution T0, 0,00X. Corollary 4.12'. Let 01jAA A be a given qua-ternion matrix (wheremnsAC ,0,1s). If rank 0nA and rank 1nA, then the quaternion sys-tem of homogeneous linear equations 0AX only exists complex solutions. Theorem 4.13. Let 01jAA A be a given quater-nion matrix, 01jXXX and 01jbb b be qua-ternion vectors (where mnsAC, 1nsXC, T12,,,ssssnbb bb, stbC, 0,1s, 1, 2,,tn). If there is at least one 00,1s such that rank 0sA  rank00ssAb, then the quaternion system of linear equations AXb has no solution. Proof. By the new definition of quaternion matrix multiplication, the quaternion system of linear equations AXb is equivalent to the system of linear equa- tions 00 011 1bXbAAX, since  000rank ra