On Bicomplex Representation Methods and Applications of Matrices over Quaternionic Division Algebra*

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 concept 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.

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 conclusions 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 introduced 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 concepts.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 quaternion 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 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, m n 

C
denote the set of m n  complex matrices, m n  H denote the set of m n  quaternion matrices and T A denote the transpose matrix of A .

The Bicomplex Representation Methods of Quaternion Matrices and the Relation between Quaternion Matrices and Complex Matrices
For any quaternion matrix where

to multiply each entries of 1
A by j from right hand side.
For above reasons, we can establish a mapping relation between quaternion matrices and complex matrices as follows:s where . The set of m n  quaternion matrices is written as A and the set of image of A is written as img A .Theorem 2.1.
. Then the mapping f is a bijective mapping from A to img A .
Proof.For any entry   0 1 , A A in img A , there exists the corresponding quaternion matrix 0 A .Simultaneously, since any quaternion matrix in A can be uniquely represented as the form (2.1), so f is an in- 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.

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 For any matrix , and a  H be given.Then is called the *-product of quaternion and quaternion vector, where 0 . Now, we introduce the following concept to quaternion division algebra.Definition 3.4.For any quaternion matrix is said to be the determinant of A , where . is the determinant of a complex matrix.Note: when 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 A is a n n  quaternion matrix and i j  , then we have 1) 2) If quaternion matrix B is obtained from quaternion matrix A by interchanging two rows (or columns) of A , then   B A .
3) If quaternion matrix A has a zero row (or column), then 0  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  B A .8) Let A and B be n n  quaternion matrices re- spectively.We have    A B A B .Up to now, people still treat the inverse matrix concept of quaternion matrix as complex matrix, that is, if quaternion matrix A satisfies 1   A A E (where E is a real unit matrix), then people think that quaternion matrix 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 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 quaternion matrix.
Definition 3.5.Let be given (where 0 1 , A A both are complex matrices).If the inverse matrices of 0 A and 1 1  A both exist, then quaternion matrix A is said to be invertible and the inverse matrix is written as A .The inverse of quaternion matrix under the new definition has the same fundamental properties as those under the traditional algebra system.It is easy to show the following facts by the new concept, namely, if a quaternion matrix A is invertible, then we have: where A are all invertible quaternion matrices, then Obviously, by the new definition of inverse of quaternion 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 ,   P P P j , 0 1 , For similar quaternion matrices, we will deduce many important properties in the next section.

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 quaternion division algebra theory, so under the new concept system, we will study firstly the eigenvalues of quaternion 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 , if there exists nonzero quaternion vector and a quaternion 0 to be the left eigenvalue of A , and X is the left eigenvector 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 eigenvector respectively.
According to the new definition of quaternion matrix multiplication and   and We have We have , , , m     (where 0 A , 1 A both are complex matrices), then the left quaternion eigenvalues of matrix 0


, we know that both 0  and 1  are not zeroes.
So there are two cases as follows: 1) When  0  , obviously, we have To sum up 1), 2) and Theorem 4.1, we can draw the conclusion.
The proof is complete.□ , , , m

  
 (where 0 A , 1 A both are complex matrices), then the right quaternion eigenvalues of matrix 0 . This proof is similar toTheorem 4.3.So we omit it here.
Theorem 4.5.Let then A and T A have the same quaternion left (right) eigenvalues.
Proof.Since 0 Proof.Since  is the left quaternion eigenvalue of A , then there exits nonzero vector  such that So  is the right quaternion eigenvalue of T A , by Theorem 4.5, we know  is the right quaternion eigenvalue of A .The same proof to  .So, the proof is complete.□ Specially, when Note: By the new definition of quaternion multiplication, the left quaternion eigenvalue of a quaternion matrix is equivalent to its right quaternion eigenvalue.So they are both called quaternion eigenvalue of the quaternion matrix.
In the following, we show an important result.The proof is complete.□ Theorem 4.8 (The generalized Cayley-Hamilton theorem over quaternion division algebra).A quaternion matrix A must be the root of its characteristic polynomial According to Definition 3.4, we know that: , where   According to the Cayley-Hamilton theorem on complex field, we know   . It indicates that quaternion matrix A must be the root of its characteristic polynomial   f  .So, the proof is complete.□ Theorem 4.9.Let (where , n n   A A C ) be given.Quaternion matrix A is diagonalizable matrix if and only if the geometric multiplicity of 0 A and 1 A both equal their algebraic multiplicity 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 1 1  matrix.So we can define the inverse of quaternion as follows: Definition 4.3.For any quaternion 0 1 a a a j   , if neither of 0 a and 1 a are zeroes, then   is said to be the inverse of a , where It is easy to verify the following facts.For any , a b H  , we have: 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: is said to be quaternion polynomial with complex coefficients (where , 0,1, , x is the *-product of i quaternion x and 0 x  is unit quaternion).
Theorem 4.10.Let   f x be a quaternion polynomial with complex coefficients.Then   f x has infinite quaternion roots.
Proof.By Fundamental Theorem of algebra,   f x exists at least one complex root 0 x , then for any given complex number 1 x , obviously, 0 1 The proof is complete.□ 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 X b , where operator '  ' denotes the new multiplication of quaternion matrices.
As we known, for any and the fundamental system of solutions to the system of homogeneous linear equations s s  0 A X is respectively, then any solution to the quaternion system of homogeneous linear equations   0 A X can be expressed as follows: where , 0, , , 0,1 By the new definition of quaternion matrix multiplication, the quaternion system of homogeneous linear equations   0 A X is equivalent to the system of homogeneous linear equations 0 0 . Since any solution to the system of homogeneous linear equations s s  0 A X can expressed as and the solutions of the quaternion system of homogeneous linear equations X .So we can draw the conclusion.
So, the proof is complete.□ Corollary 4.12.Let 0 1 j   A A A be a given quaternion matrix (where , then the quaternion system of homogeneous linear equations   0 A X only exists complex solutions.
If there is at least one , then the quaternion system of linear equations   A X b has no solution.Proof.By the new definition of quaternion matrix multiplication, the quaternion system of linear equations   A X b is equivalent to the system of linear equations 0 0 0 that is, the quaternion system of linear equations   A X b has no solution.
So, the proof is complete.□ Theorem 4.14.Let 0 1 j   A A A be a given quaternion matrix and 0 1 j   X X X be a given quaternion vector (where ).We suppose that the fundamental system of solutions to the system of linear equations s is a special solution of the system of linear equations s s s A X b  respectively, and rank   , then any solution to the quaternion system of linear equations   A X b can be expressed as: Proof.By the new definition of quaternion matrix multiplication, the quaternion system of linear equations   A X b is equivalent to the system of linear equations 0 0 0 ), so any solution to the system of quaternion linear equations   A X b can be expressed as: Proof.By the new definition of quaternion matrix multiplication, the quaternion system of linear equations A X b   is equivalent to the system of linear equations

5 ) A . 6 ) 7 )
If the jth row (column) of quaternion matrix A equal a multiple of the ith row (column) of the matrix, then 0 Suppose that A , B and C are all n n  quaternion 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  If quaternion matrix B is the n n , if there exists an invertible quaternion matrix P such that     A P B P , then A and B are said to be similar quaternion matrices written as  A B .Note: when ,

1 A
said to be the characteristic polynomial of A (where the operator  denotes the determinant of quaternion matrix under Definition 3.4).both are complex matrices), if  and  are the left eigenvalues of 0  ) are the left quaternion eigenvalues of A .Proof.Since  and  are the left eigenvalues of 0 A and 1 A respectively, then there exist nonzero vectors 1 n  C

4 . 2 .Y 4 . 2 . 1 A
left quaternion eigenvalues of A .The proof is complete.□ Similarly, we introduce a new right quaternion eigenvalue concept.Definition For any matrix , then  is said to be the right quaternion 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 are both complex matrices), if  and  are the right eigenvalues of 0  ) are the right quaternion eigenvalues of A .Proof.Since  and  are the right eigenvalues of 0

4 . 3 .
quaternion eigenvalues of A .The proof is complete.□ Theorem If the left eigenvalues of 0

Theorem 4 . 4 .
If the right eigenvalues of 0 be given.If A ∽ B , then A and B have the same eigenvalues.Proof.Since A ∽ B , there exists an invertible matrix ).We know s B and s A   0,1 s  have the same eigenvalues.By Theorem 4.3 and Theorem 4.4, we can draw that A and B have the same eigenvalues.

Theorem 4 . 11 .
Let   f x be a quaternion polynomial with complex coefficients and quaternion matrix (where, both 0 A and 1 A are compex matrices).If  is the eigenvalue of 0A , then   f  is the eigenvalue of   f A .Proof.According to the new definition of quaternion multiplication, we can easily obtain system of linear equations   A X b exists unique solution.

Definition 3.3. Let
we establish a new definition as follows.
Since any solution to the system of