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.
| Published in | American Journal of Applied Mathematics (Volume 7, Issue 6) |
| DOI | 10.11648/j.ajam.20190706.11 |
| Page(s) | 152-156 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Linear Algebraic Systems, MPGI, Rank, SVD, Least Squares Solutions
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APA Style
Asmaa Mohammed Kanan, Asma Ali Elbeleze, Afaf Abubaker. (2019). Applications of the Moore-Penrose Generalized Inverse to Linear Systems of Algebraic Equations. American Journal of Applied Mathematics, 7(6), 152-156. https://doi.org/10.11648/j.ajam.20190706.11
ACS Style
Asmaa Mohammed Kanan; Asma Ali Elbeleze; Afaf Abubaker. Applications of the Moore-Penrose Generalized Inverse to Linear Systems of Algebraic Equations. Am. J. Appl. Math. 2019, 7(6), 152-156. doi: 10.11648/j.ajam.20190706.11
AMA Style
Asmaa Mohammed Kanan, Asma Ali Elbeleze, Afaf Abubaker. Applications of the Moore-Penrose Generalized Inverse to Linear Systems of Algebraic Equations. Am J Appl Math. 2019;7(6):152-156. doi: 10.11648/j.ajam.20190706.11
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Download@article{10.11648/j.ajam.20190706.11,
author = {Asmaa Mohammed Kanan and Asma Ali Elbeleze and Afaf Abubaker},
title = {Applications of the Moore-Penrose Generalized Inverse to Linear Systems of Algebraic Equations},
journal = {American Journal of Applied Mathematics},
volume = {7},
number = {6},
pages = {152-156},
doi = {10.11648/j.ajam.20190706.11},
url = {https://doi.org/10.11648/j.ajam.20190706.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20190706.11},
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.},
year = {2019}
}
TY - JOUR T1 - Applications of the Moore-Penrose Generalized Inverse to Linear Systems of Algebraic Equations AU - Asmaa Mohammed Kanan AU - Asma Ali Elbeleze AU - Afaf Abubaker Y1 - 2019/11/18 PY - 2019 N1 - https://doi.org/10.11648/j.ajam.20190706.11 DO - 10.11648/j.ajam.20190706.11 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 152 EP - 156 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20190706.11 AB - 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. VL - 7 IS - 6 ER -
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