Star-jacobian Matrix of Star with α Coefficient α

  IJMTT-book-cover
 
International Journal of Mathematics Trends and Technology (IJMTT)
 
© 2021 by IJMTT Journal
Volume-67 Issue-2
Year of Publication : 2021
Authors : Mohamed Moktar Chaffar, Hassen Ben Mohamed
  10.14445/22315373/IJMTT-V67I2P513

MLA

MLA Style: Mohamed Moktar Chaffar, Hassen Ben Mohamed. "Star-jacobian Matrix of Star with α Coefficient α" International Journal of Mathematics Trends and Technology 67.2 (2021):85-102. 

APA Style: Mohamed Moktar Chaffar, Hassen Ben Mohamed(2021). Star-jacobian Matrix of Star with α Coefficient α. International Journal of Mathematics Trends and Technology, 85-102.

Abstract
In this paper, we introduce a new data type of Star-matrices and we define a simple basis Star-Jacobian vector that enables the representation of Star-Jacobian matrices (directly, indirectly) composed of the solution of a Star-System with α coefficient. The resolution of Star-Systems is laid in a basis of the known Gauss’ method (method of exception of unknown values) for solution of system of linear equations. When we solve a linear Star-System, we will place the 5 Star-Vectors that are the solution (linearly independent) in the columns of a matrix. The so-called fundamental matrix of the Star-System (5x5). Thereafter, we start with two examples with detailed solutions are presented. This can, in particular, be exploited to obtain arithmetic properties for classes of Star-Matrices. According to a number of different studies, we also note that there is a constant coefficient matrix Cα★ if we multiply that matrix by M★+ (Star-Matrix directly). Then you will get the matrix M★- (Star-Matix indirectly). Cα★ is an orthogonal matrix (t Cα★ = (Cα★)-1). On the other hand, we study the relationship between two Star-Jacobian matrices of Star with α coefficient, a relationship refers to the correspondence between two Star-Matrices. The results of calculations show that the products between two Star- Jacobian Matices of two Star-System (★α1, ★α2) with α1 and α2 coefficient (directly-indirectly) or (directly-directly) or (indirectly-indirectly) are diagonalizable.

Reference

[1] Chaffar, Mohamed Moktar, Star with α coefficient in the set of real numbers, Journal of Applied and Computational Mathematics 9(450)(2020).
[2] Chaffar, Mohamed Moktar, Matrix representation of a star, Journal of Applied and Computational Mathematics 9(448)(2020).
[3] Ivan Kyrchel, The Theory of the Column and Row Determinants in a Quaternion Linear Algebra, Advances in mathematics Research 15(2012)
[4] Robert Vein Paul Dale, Determinants and Their Applications in Mathematical Physics, Mathematical Sciences Springer 134(2011)
[5] Ron Larson, Bruce H Edwards, Calculus, Brooks/Col, Cengage Learning (2009).
[6] Joseph (Yossi) Gil, On the arc length parametrization problem, International Journal of Pure and Applied Mathematics, 31(3/2006) 401-419.
[7] Rafael Lopez, Differential Geometry of Curves and Surfaces in Lorentz-Minkowski space, International Electronic Journal of Geometry, (2008).
[8] Kande Dickson Kinyua, Kuria Joseph Gikonyo, Differential Geometry: An Introduction to the Theory of Curves, International Journal of Theoretical and Applied Mathematics, (2017) 225-228.
[9] Aneshkumar M, Students’ Understanding of Solving a System of Linear Equations Using Matrix Methods: A Case Study. Int J Edu Sci, 21(1-3) 124-134 (2018).
[10] Suriya Gharib, Syeda Roshana Ali, Rabia Khan, Nargis Munir& Memoona Khanam, System of Linear Equations, Guassian Elimination, Global Journal of Computer Science and Technology, 15(5)(2015) Version 1.0.
[11] Sepideh Stewart, Christine Andrews-Larson, Michelle Zandieh, Linear algebra teaching and learning: themes from recent research and evolving research priorities, Springer ZDM 51(2019) 1017–1030.
[12] Guershon Harel, The learning and teaching of linear algebra: Observations and generalizations, The Journal of Mathematical Behavior, 46(2017) 69-95 .
[13] Sierpinska A, On some aspects of students’ thinking in linear algebra. In J-L. Dorier (Eds.) On the teaching of Linear Algebra. Dordrecht, Netherlands: Kluwer Academic Publishers (2000) (209-246).
[14] E.A.Rawashdeh, A simple method for finding the inverse matrix, Matematicki vesnik, (2019) 207–213.
[15] B.S.Kashin, Decomposing an orthogonal matrix into two submatrices with extremally small (2,1)-norm, Russian Mathematical Surveys, 72(5)(2017).
[16] Antonio J.Durán, F.Alberto Grünbaum, Orthogonal matrix polynomials satisfying second-order differential equations, International Mathematics Research Notices, 10(2004) 461–484.
[17] Shilpee Srivastava Saxena, Periodic Solutions of Functional Difference Equations, International Journal of Mathematics Trens and Technology, 67(2)(2021) 36-42.

Keywords : Coefficient star-matrix, algebra matrix, algebra linear equations matrix, Star-Jacobian matrix, Star.