International Journal of Mathematics Trends and Technology (IJMTT) Volume 48 Number 4 August 2017

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1 International Journal of Mathematics Trends and Technology (IJMTT) Volume 48 Number 4 August 27 On Bicomplex Representation Methods and Application of uaternion Range Hermitian matrices (q-p) S. Sridevi (), Dr.K.Gunasekaran (2) Guest lecturer (), HOD of mathematics (2) PG and Research Department of Mathematics, Government Arts College (Autonomous), Kumbakonam , Tamil nadu, India. Abstract: In this paper, a series of bicomplex representation methods of q-p matrices is introduced. We present a new multiplication of q-p matrices, a new determinant concept, a new inverse concept of q-p matrix and a new similar matrix concept. Keywords: q-p matrix, q-p determinant, Inverse of q-p matrix, similar q-p matrix. AMS Classification: 5A57, 5A5, 5A9 Introduction: Throught we shall deal with quaternion matrices[26]. Let A* denote the conjugate transpose of A. Any matrix A H nxn is called q-p(27) if R(A)=R(A * ) and his called q-p r, if A is qs-p and rk(a)=r, where N(A), R(A) and rk(a) denote the null space, range space and rank of A respectively. denote the q-p matices. It is well known that sum, sum of parallel summable q-p matrices, Product of q-p and Generalized Inverses, Group Inverses And Reverse Order Law For Range uaternion Hermitian Matrices (q-p)[28-3]. In recent years, the algebra problems over quaternion division algebra have drawn the attention of mathematics and physics researchers [-2]. uaternion algebra theory 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 [3-7] on quaternion algebra 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 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 [,3,4,6,,8] since Dieudonne firstly introduced the quaternion determinant in 943. In addition, the inverse of quaternion matrix has not been well defined 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 [9] 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 [,3,6,,,2]. However, as Wu in [9] mentioned, these concept system is not suitable for complex algebra. In this paper, based on the bicomplex form of q- P matrix, we present some new concepts to quaternion division algebra. These new concepts can perfect the theory of Wu in [9] and unify the complex algebra and quaternion division algebra. This paper is organized as follows. In Section, we introduce a complex representation method of quaternion P matrices and explore the relation between q-p matrices and complex matrices. In Section 2, we present a series of new concepts over q-p division algebra and study their properties. In section 3, we establish some important theorems to illustrate the applications and effectiveness of the new concept system. ISSN: Page 25

2 International Journal of Mathematics Trends and Technology (IJMTT) Volume 48 Number 4 August 27. The Bicomplex Representation Methods of uaternion P matrices(q-p) and the Relation between q-p matrices and complex matrices: For any q-p matrix A uniquely represented as A = A +A j, A can be R(A) = R(A +A j) (by [27],..) R(A) = R(A )+R(A )j Where A s C (s=,), A j means to multiply each entries of A by j from right hand side and rk(a ) = rk(a j). For above reasons we can establish a mapping relation between quaternion P matrices and complex matrices as follows: f : A (A,A ) (..2) where A s C (s=,). The set of quaternion matrices is written as A and the set of image of A is written as A img. Theorem. Let f:a (A,A ), (A s C (s=,), Then the mapping f is a bijective mapping from A to A img. For any entry (A,A ) in A img, there exists the corresponding q-p matrixa = A +A j in A, therefore f is a surjection from A to A img. Simultaneously, since any q-p matrix in A can be uniquely represented as the form (..), So f is an injection from A to A img. Thus f is bijective mapping from a to A img.the proof is complete. Theorem.2 Bijection f : A (A,A ), A s C (s =,) is an isomorphism mapping from A to A img. By the concept of isomorphism mapping, this theorem is easy to prove and we omit it here. 2. The Bicomplex Matrix concept System over q-p According to the complex representation of q-p matrices above, a series of new definition of quaternion division algebra which are helpful to discuss the algebra problems on quaternion division algebra can be given as follows. Definition 2. The matrix = +j is said to be an unit q-p matrix if is an unit matrix, over complex field. In particular, if n=, then R( ) = R()+R()j =R()+()j is said to be a unit q-p written as a u. Definition 2.2 Let A = A +A j and B = B +B j nxt given. The operator R(A*B) = R(A B ) + R(A B )j (where A B,A B are both the multiplications of complex matrices) is called the * product of q-p matrices A and B. In particular, if n=t=, then we can drive the * product of q-p. Note: when AC BC, then A * B=AB. Under the definition 2. and 2.2, we give some relative properties.for any matrix A,B have: be, we ) *A = A* =A, where is a q- P matrix: 2) A+B = B+A; 3) (A+B)*C = A*C+B*C; 4) (A*B) T = B T * A T ; 5) T r (A*B) = T r (B*A) ISSN: Page 25

3 International Journal of Mathematics Trends and Technology (IJMTT) Volume 48 Number 4 August 27 Similarly, we establish a new definition as follows. Definition 2.3 Let x and ah be given. Then a*x = x*a = nx a x +a x j is called the * product of quaternion P and quaternion P vector, where R(x) = R(x )+R(x j), x C nx, x C nx, a = a +a j, a C, a C. Definition 2.4 For any q-p matrix A (A=A +A j), A = A j is said to be the determinant of A, where. is the determinant of A, where. is the determinant of a complex matrix. Note: when AC, then A =A The definition 2.3 is reasonable. First of all, the result of a q-p matrix determinant under definition 2.4 is a q-p. Secondly, from 2.4 we can see that it can convert the determinant of a q-p 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 q-p matrix and i j, then we have ) R( A ) =R( A T ) 2) If quaternion P matrix B is obtained from quaternion P matrix A by interchanging two rows (or columns) of A. then R( B ) = R(- A ) 3) If quaternion P matrix A has a zero row (or column), then R( A )=R( A T )= 4) R( K*A ) =R( K *n A ), where K *n = K*K* *K, K H n 5) If the j th row (column) of quaternion matrix A equal. A multiple of the i th row(column) of the matrix A, then A = 6) Suppose that A, B and C are all q-p matrices If all rows of B and C both equal the corresponding to rows(columns) of A except that the i th row(column) of matrix A equal the sum of the i th of B and C, then R( A ) = sr ( B )+R( C ). 7) If quaternion P matrix B is the matrix resulting from adding a multiple of the i th row(column) of matrix A to the j th row (or column) of matrix A, then R( B )=R( A ). 8) Let A and B be q-p matrices respectively. We have R( A*B ) = A * B Up to now, people still treat the inverse matrix concept of q-p matrix A satisfies A - A = (where is a real unit matrix), then people think that q-p matrix A exists its inverse matrix A -. However, people pointedly ignore two questions. An issue is how to define the product of quaternion P matrices A - and A. The other one is how to make calculation of A -. It indicates that the terminology of inverse matrix does not have a clear definition in quaternion q-p theory. In the following, we shall give a new definition and specific computational method for the inverse of q-p matrix. Definition : 2.5 Let A = A +A j be given (where A,A both are complex matrices). If the inverse matrices of A and A both exist, then q-p matrix A is said to be invertible and the inverse matrix is written as R(A ) = R( A )+R( A )j, where A, A A,A respectively. denote the inverse of complex matrices Note : when AC, then A = A -. Then inverse of q-p 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 P matrix A is invertible, then we have: ) R((A ) )= R(A) ISSN: Page 252

4 International Journal of Mathematics Trends and Technology (IJMTT) Volume 48 Number 4 August 27 2) R((A ) k ) = R((A k ) ) = R( ( ) k A )+R( ( ) k A )j, where A*=A*A*.*A is product of KA which is defined in definition 2.2 3) If A,A 2,,A m are all invertible q-p matrices, then R((A *A 2 * *A m ) ) = R( A m )*R( A m )*..*R( A ) Obviously, by the new definition of inverse of q-p matrix above, people can Determine easily whether the inverse matrix of q-p matrices exists or not and calculate the inverse matrix if possible. Under the definition of inverse of q-p matrix above, a new concept of similar q-p matrices can be given as follows: Definition 2.6 Let A,B, if there exists an invertible q-p matrix P such that A=P *B*P, then A and B are said to be similar q-p matrices written as A~B. Note: When A,BC, A=P *B*P is equivalent to A = P BP, where P = P +P j,p,p C. For similar q-p matrices, we will deduce many important properties in the next section. 2. 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 eigan value is an important issue in quaternion division algebra theory. So, under the new concept of system, we will study firstly the eigen value of q-p matrix and the relation between eigen values of similar q-p matrices in detail. Before showing the application, we will introduce firstly some concepts associate with eigen value. Definition 3. For any matrix A=(a ij ), if there exists non-zero quaternion vector XH nx and a quaternion j (where, are both complex numbers such that A * X= * X, then is said to be the left eigen value of A, and X is the left eigen vector corresponding to. For the sake of distinction, we call the left eigen value and the left quaternion the left q-p eigen value and the left q-p eigen vector respectively. According to the new definition of q-p matrix multiplication and A*X = A*X, we can derive ~ that ( * A)* A=. Thus f ( ) = * A is said to be characteristic polynomial of A (where the operator. denotes the determinant of q-p matrix under definition 2.4). Theorem 3.2 A q-p matrix A = A +A j (where A, A both are complex matrices), if and are the left eigen value of A and A respectively, then aj and b+ j are the left q-p eigen value of A. Since and are the left eigen values of A and A respectively, then there exist nonzero vectors. nx C A =, A and nx C such that we have A* =(A +A j)*( +j)=a = = ( aj) *, for all ac So aj and b j value of A. The proof is complete. are all the left q-p eigen ISSN: Page 253

5 International Journal of Mathematics Trends and Technology (IJMTT) Volume 48 Number 4 August 27 Similarly, we introduce a new right q-p eigen value concept. Definition 3.3 For any matrix A=(a ij ), if there exists nonzero quaternion vectorh xn and quaternion = + j(where, are both complex numbers) such that SY*A = * Y, then is said that to be the right q-p eigenvalue of A and Y is the right q-p eigenvector corresponding to. For the eigen value of q-p matrix, we have the following theorem: Theorem 3.4 A q-p matrix A = A +A j (where A,A are both complex matrices), if and are the right q-p eigen value of A and A respectively, then aj and b+ j ( r ac, c) are the right q-p eigen values of A. Since and are the right eigen values of A and A respectively, then there exists nonzero vectors A Xn C and, A. We have Xn C such that * A = ( j )*(A +A j) = A = =( aj )*, for a c ( j)*a = (+ j) * (A +A j) = A j = j = s(b+ j So, ) * ( j), for b c aj and b j eigenvalues of A.The proof is complete. Theorem 3.5 are the right q-p If the left eigen values of A are,... k and the left eigen values of A are 2, 2,... m (where A, A both are complex matrices), then the left q-p eigen values of matrix A = A + A j are { s aj or b t j,for all ac bc, s=,,k, t=,,m}. Suppose that is arbitrary left q-p eigen value of A, then n j H, Such that A * = *, that is, A A,Since We know that both and are not zeroes.so there are two cases as follows. ) When, obviously, we have {, 2... k }.So, { aj, i,2,... k}. 2) When, obviously we have { aj, i,2,... k}. So, { b j, t,2,... m} Theorem 3.6 To sum up ),2) and theorem 3.2, we can draw the conclusion. The proof is complete. If the right eigen values of A are, 2,..., k and he right eigen values of A are, 2,..., m (where A, A both are complex matrices), then the right q-p eigen values of matrix A = A + A j are { s aj, a c, b c, s=, k, t =,.m}. or b+ t This proof is similar to theorem 3.5 so we omit it here. Theorem 3.7 Let A, then A and A T have same q- P left (right) eigen values ISSN: Page 254

6 International Journal of Mathematics Trends and Technology (IJMTT) Volume 48 Number 4 August 27 Since A=A +A j where A C,A C ), then A T =A T +A T j. We know A and A T have the same left(right) eigen values (,2), by Theorem 3.5 and 3.6, we can draw the conclusion. Theorem 3.8 Let A and, H be given. If ( ) is the left (right) q-p eigen value of A. Then ( ) is the right(left) q-p eigen value of A. Since is the left q-p eigen value of A, then there exists nonzero vector such that A* = * Then (A* )T=( * )T. we can have T T T * A *. So, is the right q-p eigen value of A T by theorem 3.7, we know is the right q-p eigen value of A. the same proof to. So, the proof is complete. Specially, when AC, if ( ) is the left(right) eigen value of A, then ( ) is the right(left) eigen value of A. Note: By the new definition of q-p multiplication, the left q-p eigen values of a q-p matrix is equivalent to its right q-p eigen value. So they are both called q-p eigen value of the q-p matrix. Theorem 3.9 Let A,B and B have the same eigen values. P be given. If A~B, then A Since A~B, there exists an invertible matrix such that A = P *B*P, that is equivalent to A = P - BP and A = P - B P (where A = A +A j, B = B +B j, P = P +P j). We know B s and A s (s=,) have the same eigen values. By theorem 3.5 and 3.6, we can draw that A and B have the same eigen values. The proof is complete. Theorem 3. (the generalized Cayley-Hamilton theorem over q-p division algebra). A q-p matrix A must be the root of its characteristic polynomial f ( ) = * A. According to definition 2.4, we know that f ( ) = f( j) = ~ * A = ( j ) ( A Aj ) = A + A j= g( ) h( ) j Where g( ) = A, h( ) = A. According to the Cayley-Hamilton Theorem on complex field, we know g(a )=, h(a ) =. So f(a) = g(a ) + h(a ) =. It indicates that q-p matrix A must be the root of its characteristic polynomial f( ). So, the proof is complete. Theorem 3. Let A = A +A j, A, A given. A is a diagonalizable matrix if and only if both A and A are diagonalizable matrices. A is diagonalizable matrix, that is, there exists an invertible q-p matrix P such that A=P * *P. It is equivalent to A = P - P and A = P - P (where = + j is diagonal matrix). So, A is diagonalizable matrix if and only if both A and A are diagonalizable matrices, the proof is complete. be ISSN: Page 255

7 International Journal of Mathematics Trends and Technology (IJMTT) Volume 48 Number 4 August 27 Corollary 3.2 Let A = A +A j (where A,A C ) be given. If A and A both have n different eigen values, then A is diagonalizable matrix. Corollary 3.3 Let A = A +A j (wherea,a C ) be given. A q-p matrix A is diagonalizable matrix if and only if and A both have n linearly independent eigenvectors. Corollary 3.4 LetA=A +A j (wherea,a C ) be given. q-p matrix A is diagonalizable matrix if and only if the geometric multiplicity of A and A both equal their algebraic multiplicity respectively. In section 2, we have given the new definition of the inverse of q-p matrix, but that of q- P is not defined. In fact, a quaternion can be treats as a x matrix. So, we can define the inverse of quaternion as follows. Definition 3.5 For any q-p a = a +a j, if neither of a and a are zeroes, then a = a - +a - j is said to be the inverse of a, where a s - (s=,) is the reciprocal of a s. It is easy to verify the following facts. For any a,bh, we have: ) a u *a = a*a u =a; 2) a+b = b+a; 3) (a+b)*c = a*c + b*c 4) a n = (a ) n +(a ) n j 5) If a = a +a j has the inverse a, then a*a =a u In addition, we discover that there are some special phenomena about the roots at q-p polynomial under the new definition of q-p multiplication. Definition 3.6 The polynomial which has the form as follows: a *x * +a *x *(n-) +..+a n- *x * +a n *x * is said to be q- P polynomial with complex coefficients (where a i, i =,, n are all complex numbers, x = x + x j,x * is the * product of i th quaternion x and x * is unit q-p. Theorem 3.7 Let f(x) be a q-p polynomial with complex coefficient. Then f(x) has infinite q-p roots. By fundamental theorem of algebra, f(x) exist at least one complex root x, then for any given complex number x, obviously, x +x j is the root of f(x).the proof is complete. Theorem 3.8 Let f(x) be a q-p polynomial with complex coefficient and A = A +A j be a given q-p matrix (where, both A and A are complex matrices). If is the eigen value of A, then f( ) is the eigen value of f(a). According to the new definition of sssq- P multiplication, we can easily obtain f(a) =f(a ). Since is the eigen value of A, So f( ) is the eigen value of f(a ). The proof is complete. Under the new concept system, we can also solve the problems of system of linear equation A*X = b, where operator * denotes the new multiplication of q-p matrices. As we known, for nay A,A can be represented uniquely as A = A +A j, where A s (s=,) are complex matrices. Let ISSN: Page 256

8 International Journal of Mathematics Trends and Technology (IJMTT) Volume 48 Number 4 August 27 x =(x +x j,x 2 +x 2 j,..,x n +x n j) T and b = (b +b j,b 2 +b 2 j,.,b n +b n j)t be nx q-p vectors, then the following theorems are valid. Theorem 3.9 Let A = A +A j be given and X=X +X j be nx q-p vector. If rank(a s ) = r s and the fundamental system of solution to the system of homogeneous linear equation A sxs = is i, i2,. i( -ri) (s=,) respectively, then any solutions to the q-p system of homogeneous linear equations A*X= can be expressed as follows X=(C +C C ( - r ) ( - r ))+ (C +C C (n- r ) (n- where Cst y r )j C, t s =, n-r s, s=,. By the new definition of q-p matrix multiplication, the quaternion a system of homogeneous linear equations A*X = is equivalent to the system of homogenous linear A X sequation. Since any solution to the system of homogenous linear equations A s X s = can be expressed as X s = (C s s +C s2 s2 +..+C s( -ro) s( -r) (where Cst s C, t s =,..n-r s, s=,) and the solution of the system of homogenous linear equations A*X = are X = X +X j, so we can draw the conclusion. So, the proof is complete. Corollary 3.2 Let A = A +A j be a given q-p matrix (where A s c, s =,) If rank (A ) = rank (A ) =n, then the q-p system of homogenous linear equations A*X= has unique solution X = = {,,,} T Corollary 3.2 Let A = A +A j be a given q-p matrix (where A s c, s=,). If rank (A ) < n and rank(a ) = n, then the q-p system of homogenous linear equations A*X= only exists complex solution. Theorem 3.22 Let A = A +A j be a given q-p matrix, X = X +X j and b = b +b j be q-p vectors. (where A s c, X s c,bs=(b s,b s2.,b sn ) T, b st c, s =,, t =,2,..n). If there is at least one s {,} such that rank (A s ) < rank(a s s ), then the q-p system of linear equation A*X = b has no solution. By the new definition of q-p matrix multiplication the q-p system of linear equations A*X=b is equivalent to the system of linear equations, since rank(a s ) < rank(a s s ), equations the system of linear have no solutions A*X = b has no solution. So, the proof is complete. Theorem 3.23 Let A = A +A j be a given q-p matrix and X = X + X j be a given q-p vector (where A s c,x s c nxt,s=,). We suppose that the fundamental system of solutions to the system of linear equations A s X s = is s, s2,.. s(n-rs) (s=,) respectively and s (s=,) is a special solution of the system of linear equations A sxs = b, respectively, and rank(a s ) = rank(a s s ) (s=,), then any solution to the q-p system of linear equations A*X=b can be expressed as X=( +C +C C (n-r) (nr))+( +C +C C (n-n) (n- ))j ISSN: Page 257

9 International Journal of Mathematics Trends and Technology (IJMTT) Volume 48 Number 4 August 27 The proof is complete. Theorem 3.24 Let A = A +A j be a given q-p matrix, X = X +X j and b=b +b j be q-p vectors (where A s C, x s C,b s c nx, (s =,). If rank(a s ) = rank(a s s ) = n(s =,), then the q-p system of linear equations A*X = b exists unique solution. By the new definition of q-p matrix multiplication, the q-p system of linear equations A*X=b is equivalent to the system of linear equations and rank(a s ) = rank(a s B s )=n, we know the system of linear equations have unique solution. So the q-p system of linear equations A*X = b exists unique solution. Corollary 3.25 Let A = A +A j be a given q-p matrix and b = b +b j be a given nx q-p vector. If rank(a s ) = rank(a s B s ) = n (s=,), then the solution of the system of equations A*X=b is X = A - * b. Corollary 3.26 Let A and b = b + b j (where b s C nx, s=,,b ) be given. Then the q-p system of linear equations A*X = b has no solution. Corollary 3.27 Let A and bc nx be given. If rank(a) = rank(a), then any solution to the q-p system of linear equations A*X = b can expressed as X = a - b+aj, where ac s. References [] L. X. Chen, Inverse Matrix and Properties of Double Determinant over uaternion TH Field, Science in China (Series A), Vol. 34, No. 5, 99, pp [2] L. X. Chen, Generalization of Cayley-Hamilton Theorem over uaternion Field, Chinese Science Bulletin, Vol. 7, No. 6, 99, pp [3] R, M. Hou, X.. Zhao and l. T. Wang, The Double Determinant of Vandermonde s Type over uaternion Field, Applied Mathematics and Mechanics, Vol. 2, No.9, 999, pp. -7. [4] L. P. Huang, The Determinants of uaternion Matrices and Their Properties, Journal of Mathematics Study, Vol. 2, 995, pp. -3. [5] J. L. Wu, L. M. Zou, X. P. Chen and S. J. Li, The stimation of igen values of Sum, Difference, and Tensor Product of Matrices over uaternion Division Algebra, Linear Algebra and its Applications, Vol. 428, 28, pp [6] T. S. Li, Properties of Double Determinant over uaternion Field, Journal of Central China Normal University, Vol., 995, 3-7. [7] B. X. Tu, Dieudonne Determinants of Matrices over adivision Ring, Journal of Fudan University, 99A, Vol., pp [8] B. X. Tu, Weak Direct Products and Weak Circular Product of Matrices over the Real uaternion Division Ring, Journal of Fudan University, Vol. 3, 99, p [9] J. L. Wu, Distribution and stimation for igenvalues of Real uaternion Matrices, Computers and Mathematics with Applications, Vol. 55, 28, pp [] B. J. Xie, Theorem and Application of Determinants Spread out of Self-Conjugated Matrix, Acta Mathematica Sinica, Vol. 5, 98, pp []. C. Zhang, Properties of Double Determinant over the uaternion Field and Its Applications, Acta MathematicaSinica, Vol. 38, No. 2, 995, pp [2] W. J. Zhuang, Inequalities of igenvalues and Singular Values for uaternion Matrices, Advances in Mathematics, Vol. 4, 988, pp [3] W. Boehm, On Cubics: A Survey, Computer Graphics and Image Processing, Vol. 9, 982, pp [4] G. Farin, Curves and Surfaces for Computer Aided Geometric Design, Academic Press, Inc., San Diego CA, 99. [5] K. Shoemake, Animating Rotation with uaternioncalculus, ACM SIGGRAPH, 987, Course Notes, Computer Animation: 3 D Motion, Specification, and Control. [6]. G. Wang, uaternion Transformation and Its Application to the Displacement Analysis of Spatial Mechanisms, Acta Mathematica Sinica, Vol. 5, No., 983, pp [7] G. S. Zhang, Commutativity of Composition for Finite Rotation of a Rigid Body, Acta Mechanica Sinica, Vol. 4, 982. [8]. T. Browne, The Characteristic Roots of a Matrix, Bulletin of the American Mathematical Society, Vol. 36, 93, pp [9] J. L. Wu and Y. Wang, A New Representation Theory and Some Methods on uaternion Division Algebra, Journal of Algebra, Vol. 4, No. 2, 29, pp [2]. W. Wang, The General Solution to a System of Real uaternion Matrix quation, Computer and Mathematicswith Applications, Vol. 49, 25, pp [2] G. B. Price, An Introduction to Multicomplex Spaces and Functions, Marcel Dekker, New York, 99. [22] D. Rochon, A Bicomplex Riemann Zeta Function, Tokyo Journal of Mathematics, Vol. 27, No. 2, 24, pp [23] S. P. Goyal and G. Ritu, The Bicomplex Hurwitz Zeta function, The South ast Asian Journal of Mathematics and Mathematical Sciences, 26. [24] S. P. Goyal, T. Mathur and G. Ritu, Bicomplex Gamma and Beta Function, Journal of Raj. Academy PhysicalSciences, Vol. 5, No., 26, pp [25] J. N. Fan, Determinants and Multiplicative Functionals on uaternion Matrices, Linear Algebra and Its Applications, Vol. 369, 23, pp [26]. Zhang.F, uarternions and Matrices of quaternions, linear Algebra and its application, 25 (997), ISSN: Page 258

10 International Journal of Mathematics Trends and Technology (IJMTT) Volume 48 Number 4 August 27 [27]. Gunasekaran.K and Sridevi.S On Range uaternion Hermitian Matrices;Inter;J;Math.,Archieve-6(8), [28]. Gunasekaran.K and Sridevi.S On Sums of Range uaternion Hermitian Matrices; Inter;J.Modern ngineering Research -.5, ISS (25). [29]. Gunasekaran.K and Sridevi.S On Product of Range uaternion Hermitian Matrices;Inter.,Journal of pure algebra 6(6),26, [3]. Gunasekaran.K and Sridevi.S Generalized inverses, Group inverses and Revers order law of range quaternion matrices.,iosr- JMVol.,2, Iss.4 ver.ii(jul Aug 26),5-55. [3]. Junliang Wu and Pingping Zhang On Bicomplex Representation Methods and Applications of Matrices over uaternionic Division Algebra, Advances in Pure Mathematics, 2,, 9-5. ISSN: Page 259