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.
| Published in | Pure and Applied Mathematics Journal (Volume 5, Issue 4) |
| DOI | 10.11648/j.pamj.20160504.15 |
| Page(s) | 113-119 |
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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. |
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QR Method, Real Matrix, Eigen Value
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| [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. |
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| [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. |
APA Style
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
ACS Style
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
AMA Style
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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Download@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://article.sciencepublishinggroup.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}
}
TY - JOUR
T1 - The QR Method for Determining All Eigenvalues of Real Square Matrices
AU - Eyaya Fekadie Anley
Y1 - 2016/07/23
PY - 2016
N1 - https://doi.org/10.11648/j.pamj.20160504.15
DO - 10.11648/j.pamj.20160504.15
T2 - Pure and Applied Mathematics Journal
JF - Pure and Applied Mathematics Journal
JO - Pure and Applied Mathematics Journal
SP - 113
EP - 119
PB - Science Publishing Group
SN - 2326-9812
UR - https://doi.org/10.11648/j.pamj.20160504.15
AB - 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.
VL - 5
IS - 4
ER -
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