In 1843, the Irish mathematician (Hamilton, 1853) defined quaternions as a new number system, a generalization of complex numbers. Hamilton incorporated a new multiplication operation into the vector algebra where the partition is also possible by defining quaternions. Thus, the examination of movements is facilitated in the three-dimensional space. That is, quaternions are division algebra (Hacısalihoğlu, 1983). The other structures with division algebra are real numbers, complex numbers and octanions. (Majernik, 2006: 9-13) studied dual quaternions, which are defined as a new type of quaternions. Besides, real and binary (hyperbolic) quaternions are studied in his work. Dual quaternions can solve many problems in mathematical physics. Dual quaternions are a newly developing work and one of the first studies of this area was done by (Yüce and Ercan, 2011: 142-146). In their study, matrix representation of dual quaternions are expressed. Moreover, Euler and De-Moivre formulas for complex numbers and quaternions are generalized for dual quaternions. The set of dual quaternions are denoted by H_D and the basic operations on this set are examined. The most basic feature of this set is that the dual quaternion multiplication is commutative although the real quaternion multiplication is not commutative. Therefore, the left (right) matrix representations are the same in dual quaternions. When the literature is examined in a wide scale, no study has been found on dual quaternion matrices. In this study, matrices whose elements are dual quaternion are called "Dual Quaternion Matrix" and the set of all mxn matrices with dual quaternion coefficients is denoted by M_mxn(H_D). If m=n, then the set of dual quaternion matrices is denoted by M_n(H_D). In the continuation of the work, algebraic structures of dual quaternion matrices are investigated. Then, the basic properties of dual quaternion matrices (addition, multiplication, conjugation, transpose, conjugate transpose and trace) are examined. Moreover, the theorem that can find the power of the dual quaternion matrix is obtained. In addition, real matrix representation of dual quaternion matrices are introduced and their properties are described. Purpose: This study was carried out with the purpose of contributing to the literature by defining dual quaternion matrices and basic operations on these matrices, finding real matrix representation of these matrices and using Matlab applications. Scope: The scope of this work consists of algebraic structures of dual quaternion matrices, basic operations on these matrices and real matrix representations of dual quaternion matrices. Limitations: From the basic operations on the set of dual quaternion matrices, properties such as addition, multiplication, conjugate, transpose, conjugate transpose, trace and power are investigated. Method: Matlab (Matrix Laboratory) programming language was used in this study. Real matrix representations of dual quaternion matrices were also obtained using the Matlab program. In addition, a method for calculating the power of dual quaternion matrices is found, and also Matlab command that can calculate the power of dual quaternion matrices is written. Findings: The left (right) matrix representations of the dual quaternion matrices are the same, because the dual quaternions are commutative. The power of the dual quaternion matrix can be calculated in the Matlab program using the obtained method. Conclusion: At the end of the work, we showed that the set of dual quaternion matrices M_n(H_D) is a n^2 - dimensional module, 4 - dimensional module and 4n^2 - dimensional vector space over the dual quartenion ring H_D, the real matrix ring M_n(R) and R, respectively. Sample applications are made in terms of reinforcement of this subject, and the solution of samples are facilitated with the help of the Matlab program. It has been seen that very high order powers of dual quaternion matrices can easily be obtained with Matlab. The command written in Matlab program which finds the real matrix representations of the dual quaternion matrices is more useful because it is difficult to study on the dual quaternion matrices when the order of matrices increases. In the future researches, new contributions can be made to the dual quaternion matrices using the real matrix representations.
Anahtar Kelimeler: Dual Quaternions, Dual Quaternion Matrices, Real Matrix Representation