Pure and Applied Mathematics Journal

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The QR Method for Determining All Eigenvalues of Real Square Matrices

Received: 23 November 2015    Accepted: 03 December 2015    Published: 23 July 2016
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Abstract

Eigenvalues are special sets of scalars associated with a given matrix. In other words for a given matrix A, if there exist a non-zero vector V such that, AV= λV for some scalar λ, then λ is called the eigenvalue of matrix A with corresponding eigenvector V. The set of all nxm matrices over a field F is denoted by Mnm (F). If m = n, then the matrices are square, and which is denoted by Mn (F). We omit the field F = C and in this case we simply write Mnm or Mn as appropriate. Each square matrix AϵMnm has a value in R associated with it and it is called its determinant which is use full for solving a system of linear equation and it is denoted by det (A). Consider a square matrix AϵMn with eigenvalues λ, and then by definition the eigenvectors of A satisfy the equation, AV = λV, where v={v1, v2, v3…………vn}. That is, AV=λV is equivalent to the homogeneous system of linear equation (A-λI) v=0. This homogeneous system can be written compactly as (A-λI) V = 0 and from Cramer’s rule, we know that a linear system of equation has a non-trivial solution if and only if its determinant is zero, so the solution λ is given by det (A-λI) =0. This is called the characteristic equation of matrix A and the left hand side of the characteristic equation is the characteristic polynomial whose roots are equals to λ.

DOI 10.11648/j.pamj.20160504.15
Published in Pure and Applied Mathematics Journal (Volume 5, Issue 4, August 2016)
Page(s) 113-119
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

Keywords

QR Method, Real Matrix, Eigen Value

References
[1] Bronson, Rchard. (1991). Matrix Methods: An introduction. 2nd ed. San Diego: Academic press, inc.
[2] Faires, J. D and R. L. burden., (2002). Numerical Methods 3rd ed. Vol. 2. Pblisher: Broks cole
[3] Horn, R. A. and C. A. Johnson., (1985). Matrix Analysis. 1st ed. Cambridge: Cambridge University.
[4] H. R. saxena. (2000). Finite Difference & Numerical Analysis. Pub S. Chad company LTD
[5] Iyenger S. R. K, jain R. K. (2009), Numerical Methods, New delhi: New age international publishers.
[6] Kres. R. (1998), Graduate Texts In Mathematics, New York: spriner-verlag.
[7] Michelles. S. (1990). A simple proof of convergence of the QR Algorithm for Normal matrices without shifts. IMA Ppreprint series NO 720.
[8] Muzafar F. Hama. (2010). A Technique to Have a Convergence for the QR Algorithm.
[9] International Journal of Algebra, Vol. 6, 2012, no. 2, 65 - 72, University of Sulaimani, College of Science Department of Mathematics, Sulaimani, Iraqhamamuzafar@yahoo.com
[10] Paul Schmitz. (2012). The QR algorithm senior seminar, university of Minnesota Morris spring.
[11] Watkins, Davis S, (2008). The QR algorithm Revised. SIAM Review 50. 1: 133-145.
Author Information
  • College of Natural Science, Department of Mathematics, Arba Minch University, Arba Minch, Ethiopia

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    Eyaya Fekadie Anley. (2016). The QR Method for Determining All Eigenvalues of Real Square Matrices. Pure and Applied Mathematics Journal, 5(4), 113-119. https://doi.org/10.11648/j.pamj.20160504.15

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    Eyaya Fekadie Anley. The QR Method for Determining All Eigenvalues of Real Square Matrices. Pure Appl. Math. J. 2016, 5(4), 113-119. doi: 10.11648/j.pamj.20160504.15

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    Eyaya Fekadie Anley. The QR Method for Determining All Eigenvalues of Real Square Matrices. Pure Appl Math J. 2016;5(4):113-119. doi: 10.11648/j.pamj.20160504.15

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  • @article{10.11648/j.pamj.20160504.15,
      author = {Eyaya Fekadie Anley},
      title = {The QR Method for Determining All Eigenvalues of Real Square Matrices},
      journal = {Pure and Applied Mathematics Journal},
      volume = {5},
      number = {4},
      pages = {113-119},
      doi = {10.11648/j.pamj.20160504.15},
      url = {https://doi.org/10.11648/j.pamj.20160504.15},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.pamj.20160504.15},
      abstract = {Eigenvalues are special sets of scalars associated with a given matrix. In other words for a given matrix A, if there exist a non-zero vector V such that, AV= λV for some scalar λ, then λ is called the eigenvalue of matrix A with corresponding eigenvector V. The set of all nxm matrices over a field F is denoted by Mnm (F). If m = n, then the matrices are square, and which is denoted by Mn (F). We omit the field F = C and in this case we simply write Mnm or Mn as appropriate. Each square matrix AϵMnm has a value in R associated with it and it is called its determinant which is use full for solving a system of linear equation and it is denoted by det (A). Consider a square matrix AϵMn with eigenvalues λ, and then by definition the eigenvectors of A satisfy the equation, AV = λV, where v={v1, v2, v3…………vn}. That is, AV=λV is equivalent to the homogeneous system of linear equation (A-λI) v=0. This homogeneous system can be written compactly as (A-λI) V = 0 and from Cramer’s rule, we know that a linear system of equation has a non-trivial solution if and only if its determinant is zero, so the solution λ is given by det (A-λI) =0. This is called the characteristic equation of matrix A and the left hand side of the characteristic equation is the characteristic polynomial whose roots are equals to λ.},
     year = {2016}
    }
    

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