Some New Properties of The Real Quaternion Matrices and Matlab Applications

Kemal Gökhan Nalbant; Salim Yüce

doi:10.17776/csj.425691

TR EN

Some New Properties of The Real Quaternion Matrices and Matlab Applications

Abstract

In this study, firstly, it was shown that the set of real quaternion matrices is a -dimensional module over the real matrix ring and -dimensional module over the complex matrix ring . Moreover, some new properties of the real quaternion matrices were described. Then, matrix representations of the real quaternion matrices were found easily by Matlab. These matrices were also applied to find the inverse of the real quaternion matrices and inverse matrices were obtained easily with these matrices. In addition, some new properties for matrix representations of the real quaternion matrices were found. Also, the inverse of the real quaternion block matrices was obtained by new methods. Finally, a new method to calculate the determinant of the real quaternion matrices was found and the determinant of these matrices was calculated easily with Matlab application.

Keywords

Quaternions,real quaternions,real quaternion matrices,real matrix representation,determinant,block matrices,block matrices

References

[1]. Hamilton W.R., Elements of Quaternions, London: Longmans, Green & Company, 1866.
[2]. Brenner J.L., Matrices of Quaternions, Pacific Journal of Mathematics, 1 (1951) 329-335.
[3]. Erdoğdu M. and Özdemir M., On Complex Split Quaternion Matrices, Advances in Applied Clifford Algebras, 23-3 (2013) 625-638.
[4]. Erdoğdu M. and Özdemir M., Split Quaternion Matrix Representation of Dual Split Quaternions and Their Matrices, Advances in Applied Clifford Algebras, 25-4 (2015) 787-798.
[5]. Kösal H. H. and Tosun, M., Commutative Quaternion Matrices, Advances in Applied Clifford Algebras, 24-3 (2014) 769-779.
[6]. Wiegmann N.A., Some Theorems on Matrices with Real Quaternion Elements, Canadian Journal of Mathematics, 7 (1955) 191-201.
[7]. Wolf L.A., Similarity of Matrices in which the Elements are Real Quaternions, Bulletin of the American Mathematical Society, 42-10 (1936) 737-743.
[8]. Zhang F., Quaternions and Matrices of Quaternions, Linear Algebra and Its Applications, 251 (1997) 21-57.

[9]. Morais J.P., Georgiev S. and Sprößig W., Real Quaternionic Calculus Handbook, Birkhäuser, Springer Basel, 2014.
[10]. Huang L. and So W., On Left Eigenvalues of a Quaternionic Matrix, Linear Algebra and Its Applications, 323 (2001) 105-116.
[11]. Zhang F., Geršgorin Type Theorems for Quaternionic Matrices, Linear Algebra and Its Applications, 424 (2007) 139-153.
[12]. Cohen N. and De Leo S., The Quaternionic Determinant, Electronic Journal of Linear Algebra, 7 (2000) 100-111.
[13]. Aslaksen H., Quaternionic Determinants, The Mathematical Intelligencer, 18-3 (1996) 57-65.
[14]. Gelfand I., Retakh V. and Wilson R.L., Quaternionic Quasideterminants and Determinants, Translations of the American Mathematical Society-Series 2, 210 (2003) 111-124.
[15]. Bagazgoitia A., A Determinantal Identity for Quaternions, In Proceedings of 1983 Conference on Algebra Lineal y Aplicaciones, Vitoria-Gasteiz, Spain, (1984) 127-132.
[16]. Lewis D., A Determinantal Identity for Skewfields, Linear algebra and its applications, 71 (1985) 213-217.
[17]. Jiang T. S. and Wei M. S., On a Solution of the Quaternion Matrix Equation and Its Application, Acta Mathematica Sinica, 21-3 (2005) 483-490.
[18]. Song C., Feng J.-e, Wang X. and Zhao J., A Real Representation Method for Solving Yakubovich-j-Conjugate Quaternion Matrix Equation, Abstract and Applied Analysis, Hindawi, (2014).
[19]. Tian Y., Universal Factorization Equalities for Quaternion Matrices and Their Applications, Mathematical Journal of Okayama University, 41 (1999) 45-62.
[20]. Rodman L., Topics in Quaternion Linear Algebra, Princeton: Princeton University Press, 2014.
[21]. Lin Y. and Wang Q.-W., Completing a Block Matrix of Real Quaternions with a Partial Specified Inverse, Journal of Applied Mathematics, (2013).
[22]. Al-Zhour Z., Some New Linear Representations of Matrix Quaternions with Some Applications, Journal of King Saud University-Science, (2017).
[23]. Ahmad S.S. and Ali I., Bounds for Eigenvalues of Matrix Polynomials over Quaternion Division Algebra, Advances in Applied Clifford Algebras, 26-4 (2016) 1095-1125.
[24]. Song G. and Zhou Y., Block Independence in Various Generalized Inverses of Partitioned Quaternion Matrices, Iranian Journal of Science and Technology, Transactions A: Science, (2018) 1-10.
[25]. Hamilton W.R., Lectures on Quaternions, Dublin: Hodges and Smith, 1853.
[26]. Ward J., Quaternions and Cayley Numbers: Algebra and Applications, Mathematics and Its Applications, Dordrecht: Kluwer, 1997. [27]. Powell P.D., Calculating Determinants of Block Matrices, arXiv:1112.4379 (2011).
[28]. Silvester J.R., Determinants of Block Matrices, The Mathematical Gazette, 84 (2000) 460-467.
[29]. Tian Y. and Takane Y., More on Generalized Inverses of Partitioned Matrices with Banachiewicz-Schur forms, Linear Algebra and its Applications, 430 (2009) 1641-1655.
[30]. Meyer Jr C.D., Generalized Inverses and Ranks of Block Matrices, SIAM Journal on Applied Mathematics, 25-4 (1973) 597-602.
[31]. Moors E., On the Reciprocal of the General Algebraic Matrix, Bull. Amer. Math. Soc., 26 (1920) 394-395.
[32]. Penrose R., A Generalized Inverse for Matrices, Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, 51 (1955) 406-413.
[33]. Zhang F., The Schur Complement and Its Applications, Springer Science & Business Media, 2006.
[34]. Banachiewicz T., Zur Berechnung der Determinanten, wie auch der Inversen und zur darauf basierten Auflosung der Systeme linearer Gleichungen, Acta Astronom. Ser. C, 3 (1937) 41-67.
[35]. Frazer R.A., Duncan W.J. and Collar A. R., Elementary Matrices and Some Applications to Dynamics and Differential Equations, Cambridge: Cambridge University Press, 1938.
[36]. Gallier J., The Schur Complement and Symmetric Positive Semidefinite (and definite) Matrices, Penn Engineering, (2010).
[37]. Horn R.A. and Johnson C.R., Matrix Analysis, Cambridge: Cambridge University Press, 1990.
[38]. De Leo S., Scolarici G. and Solombrino L., Quaternionic Eigenvalue Problem, Journal of Mathematical Physics, 43-11 (2002) 5815-5829.
[39]. Jiang T. and Ling S., On a Solution of the Quaternion Matrix Equation and Its Applications, Advances in Applied Clifford Algebras, 23-3 (2013) 689-699.

Details

Primary Language

English

Subjects

-

Journal Section

Research Article

Authors

Kemal Gökhan Nalbant ^*
0000-0002-5065-2504
Türkiye

Salim Yüce
0000-0002-8296-6495
Türkiye

Publication Date

March 22, 2019

Submission Date

May 21, 2018

Acceptance Date

January 2, 2019

Published in Issue

Year 2019 Volume: 40 Number: 1

DOI

https://doi.org/10.17776/csj.425691

IZ

https://izlik.org/JA46FE84CA

Cite

RIS / Bibtex

APA

Nalbant, K. G., & Yüce, S. (2019). Some New Properties of The Real Quaternion Matrices and Matlab Applications. Cumhuriyet Science Journal, 40(1), 42-60. https://doi.org/10.17776/csj.425691

Some New Properties of The Real Quaternion Matrices and Matlab Applications

Reel Kuaterniyon Matrislerinin Bazı Yeni Özellikleri ve Matlab Uygulamaları

Öz

Anahtar Kelimeler

Some New Properties of The Real Quaternion Matrices and Matlab Applications

Abstract

Keywords

References

Details

Primary Language

Subjects

Journal Section

Authors

Publication Date

Submission Date

Acceptance Date

Published in Issue

DOI

IZ

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