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Reel Kuaterniyon Matrislerinin Bazı Yeni Özellikleri ve Matlab Uygulamaları

Year 2019, Volume: 40 Issue: 1, 42 - 60, 22.03.2019

Kemal Gökhan Nalbant , Salim Yüce

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

Abstract

Bu çalışmada, ilk olarak,  reel kuaterniyon
matrislerin kümesinin  reel matris halkası
üzerinde boyutlu bir modül olduğu
ve kompleks matris halkası üzerinde  boyutlu bir modül olduğu
gösterilmiştir. Ayrıca, reel kuaterniyon matrislerin bazı yeni özellikleri
tanımlanmıştır. Daha sonra, reel kuaterniyon matrislerin matris temsilleri
Matlab uygulamaları ile kolayca elde edilmiştir. Bu matrisler reel kuaterniyon
matrislerin tersini bulmak için de uygulanmış ve bu matrislerle ters matrisler
kolaylıkla elde edilmiştir. Buna ek olarak, reel kuaterniyon matrislerin matris
temsilleri için bazı yeni özellikler bulunmuştur. Ayrıca, tipindeki reel kuaterniyon blok matrislerin tersi yeni
yöntemlerle elde edilmiştir. Son olarak,  tipindeki reel
kuaterniyon matrislerin determinantını hesaplamak için yeni bir yöntem bulunmuş
ve Matlab uygulaması ile bu matrislerin determinantı kolayca hesaplanmıştır.

Keywords

Kuaterniyonlar reel kuaterniyonlar reel kuaterniyon matrisler reel matris temsili determinant blok matrisler

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.

Some New Properties of The Real Quaternion Matrices and Matlab Applications

Year 2019, Volume: 40 Issue: 1, 42 - 60, 22.03.2019

Kemal Gökhan Nalbant , Salim Yüce

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

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.
There are 38 citations in total.

Details

Primary Language English
Journal Section Natural Sciences
Authors

Kemal Gökhan Nalbant 0000-0002-5065-2504

Salim Yüce 0000-0002-8296-6495

Publication Date March 22, 2019
Submission Date May 21, 2018
Acceptance Date January 2, 2019
Published in Issue Year 2019Volume: 40 Issue: 1

Cite

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