Volume 7, Issue 6, December 2019, Page: 152-156
Applications of the Moore-Penrose Generalized Inverse to Linear Systems of Algebraic Equations
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:
Received: Oct. 4, 2019;       Accepted: Oct. 30, 2019;       Published: Nov. 18, 2019
DOI: 10.11648/j.ajam.20190706.11      View  555      Downloads  207
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.
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 © 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.
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