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Applications of the Moore-Penrose Generalized Inverse to Linear Systems of Algebraic Equations

Received: 4 October 2019     Accepted: 30 October 2019     Published: 18 November 2019
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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.

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), 2019. Published by Science Publishing Group

Keywords

Linear Algebraic Systems, MPGI, Rank, SVD, Least Squares Solutions

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.
Cite This Article
  • 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

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    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

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    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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  • @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}
    }
    

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    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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Author Information
  • Mathematics Department, Science Faculty, Sabratha University, Sabratha, Libya

  • Mathematics Department, Science Faculty, Zawia University, Zawia, Libya Email address:

  • Mathematics Department, Science Faculty, Zawia University, Zawia, Libya Email address:

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