Linear Equations Systems of Real and Complex Semi-Quaternions
- PDF / 319,402 Bytes
- 11 Pages / 595.276 x 790.866 pts Page_size
- 49 Downloads / 214 Views
(0123456789().,-volV)(0123456789(). ,- volV)
RESEARCH PAPER
Linear Equations Systems of Real and Complex Semi-Quaternions Yasemin Alago¨z1 • Go¨zde O¨zyurt1 Received: 4 November 2019 / Accepted: 3 August 2020 Ó Shiraz University 2020
Abstract In this work firstly real semi-quaternion matrices and their properties are examined. Then, 2n 2n complex adjoint matrix and 4n 4n real matrix representation of a real semi-quaternion matrix are expressed. Further systems of linear real semiquaternionic equations are investigated and in some of them the 2n 2n complex adjoint matrix and the 4n 4n real matrix representation are used. Moreover, matrices which entries are complex semi-quaternions and their 2n 2n real semi-quaternion matrix representation are presented. Additionally system of linear complex semi-quaternionic equations is described with this 2n 2n matrix representation. Finally, for a complex semi-quaternion matrix 4n 4n complex adjoint matrix is introduced. Also, linear complex semi-quaternionic equations systems are examined. Keywords Real semi-quaternion Complex semi-quaternion Real semi-quaternion matrix Complex semi-quaternion matrix Linear real semi-quaternionic equations system Linear complex semi-quaternionic equations system Mathematics Subject Classification 15B33 15A06
1 Introduction Quaternions have considerable applications to many areas of mathematics such as algebra, analysis, geometry and also widely use in computer graphics, signal processing, altitude control, physics and mechanics (Kuipers 1999; Vince 2011; Morais et al. 2014). The main difficulty in the study of quaternion matrices is the noncommutativity multiplication of quaternions, so it is often worked on a quaternion matrix by dealing with a pair of complex matrix. Wolf (1936) gave similarity of two matrices which the elements are real quaternions. Lee (1949) defined an isomorphism between quaternion matrices and complex matrices and thus examined the eigenvalues of noncommutative quaternion matrices using these complex matrices. Brenner (1951) and Weigmann (1955) showed that some theorems hold for complex matrices are also hold for quaternion matrices. By means of complex
¨ zyurt & Go¨zde O [email protected] Yasemin Alago¨z [email protected] 1
Department of Mathematics, Yildiz Technical University, Istanbul, Turkey
representation, Zhang (1997) studied quaternion matrices and gave new proofs for certain known results. Jiang (2004) examined the problems of quaternionic linear equations and presented an algorithm for quaternionic linear equations in quaternionic quantum theory. Ward (1997) expressed 4 4 real matrix representations of quaternions. Furthermore, approaches of quaternions in terms of 2 2 complex matrices and 4 4 real matrices allow describing spin in quantum mechanics, rotations in 3and 4-dimensional space (Morais et al. 2014; Ward 1997). Bidimensional (2D) signals by means of quaternion matrices also uses for vector–sensor signal modeling and processing (Le Bihan and Mars 2004). Jia
Data Loading...
