Applications of the Moore-Penrose Generalized Inverse to Linear Systems of Algebraic Equations
American Journal of Applied Mathematics
Volume 7, Issue 6, December 2019, Pages: 152-156
Received: Oct. 4, 2019; Accepted: Oct. 30, 2019; Published: Nov. 18, 2019
Views 22      Downloads 27
Authors
Asmaa Mohammed Kanan, Mathematics Department, Science Faculty, Sabratha University, Sabratha, Libya
Asma Ali Elbeleze, Mathematics Department, Science Faculty, Zawia University, Zawia, Libya Email address:
Afaf Abubaker, Mathematics Department, Science Faculty, Zawia University, Zawia, Libya Email address:
Article Tools
Follow on us
Abstract
In this work, we consider linear systems of algebraic equations. These systems are studied utilizing the theory of the Moore-Penrose generalized inverse or shortly (MPGI) of matrices. Some important algorithms and theorems for computation the MPGI of matrices are given. The singular value decomposition (SVD) of a matrix has a very important role in computation the MPGI, hence it is useful to study the solutions of over- and under-determined linear systems. We use the MPGI of matrices to solve linear systems of algebraic equations when the coefficients matrix is singular or rectangular. The relationship between the MPGI and the minimal least squares solutions to the linear system is expressed by theorem. The solution of the linear system using the MPGI is often an approximate unique solution, but for some cases we can get an exact unique solution. We treat the linear algebraic system as an algebraic equation with coefficients matrix A (square or rectangular) with complex entries. A closed form for solution of linear system of algebraic equations is given when the coefficients matrix is of full rank or is not of full rank, singular square matrix or non-square matrix. The results are taken from the works mentioned in the references. A few examples including linear systems with coefficients matrix of full rank and not of full rank are provided to show our studding.
Keywords
Linear Algebraic Systems, MPGI, Rank, SVD, Least Squares Solutions
To cite this article
Asmaa Mohammed Kanan, Asma Ali Elbeleze, Afaf Abubaker, Applications of the Moore-Penrose Generalized Inverse to Linear Systems of Algebraic Equations, American Journal of Applied Mathematics. Vol. 7, No. 6, 2019, pp. 152-156. doi: 10.11648/j.ajam.20190706.11
Copyright
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]
S. L. Campbell, C. D. Meyer, Jr., Generalized Inverses of Linear Transformations, Pitman, London, 1979.
[2]
W. W. Hager, Applied Numerical Linear Algebra, Prentice Hall, USA, 1988.
[3]
A. Ben-Israel, T. N. E. Greville, Generalized Inverses: Theory and Applications, Springer-Verlage, New York (2003).
[4]
G. H. Golub, Least squares, Singular values and matrix approximations, Aplikace matematiky, 13 (1968) 44-51.
[5]
G. H. Golub, C. Reinsch, Singular value decomposition and least squares solutions, Numer. Math., 14 (1970) 403-420.
[6]
G. Bahadur-Thapa, P. Lam-Estrada, J. Lo`pez-Bonilla, On the Moore-Penrose Generalized Inverse Matrix, World Scientific News, 95 (2018) 100-110.
[7]
M. Zuhair Nashed (Ed.), Generalized inverses and applications, Academic Press, New York, (1976).
[8]
R. Penrose, A generalized inverse for matrices, Proc. Camb. Phil. Soc., 51 (1955) 406-413.
[9]
H. Ozden, A Note On The Use Of Generalized Inverse Of Matrices In Statistic, Istanbul Univ. Fen Fak. Mat. Der., 49 (1990) 39-43.
[10]
A. Ben-Israel, Generalized Inverses of Matrices and Their Applications, Springer, (1980) 154-186.
[11]
C. E. Langenhop, On Generalized Inverse of Matrices, SIAM J. Appl. Math., 15 (1967) 1239-1246.
[12]
C. R. Rao, S. K. Mitra, Generalized Inverse of a Matrix and Its Applications, New York: Wiley, (1971) 601-620.
[13]
A. M. Kanan, Solving the Systems of Linear Equations Using the Moore-Penrose Generalized Inverse, Journal Massarat Elmeya, Part 1 (2) (2017) 3-8.
[14]
J. Lopez-Bonilla, R. Lopez-vazquez, S. Vidal-Beltran, Moore-Penrose's inverse and solutions of linear systems, World Scientific News, 101 (2018) 246-252.
[15]
Abu-Saman, A. M., Solution of Linearly-Dependent Equations by Generalized Inverse of Matrices, Int. J. Sci. Emerging Tech, 4 (2012) 138-142.
[16]
J. Z. Hearon, Generalized inverses and solutions of linear systems, Journal of research of notional Bureau of standards-B. Mathematical Sciences, 72B (1968) 303-308.
[17]
D. H. Griffel, Linear Algebra and its Applications, vol. 1, first course.
[18]
F. Chatelin, Eigenvalues of Matrices, WILEY, France, 1807.
[19]
T. N. E. Greville, The Pseudoinverse of a rectangular singular matrix and its application to the solution of systems of linear equations, SIAM Rev., 1 (1960) 38-43.
[20]
G. H. Golub, W. Kahan, Calculating the singular values and pseudo-inverse of a matrix, SIAM J. Numer. Anal., 2 (B) (1965) 205-224.
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186