Archive
Special Issues Volume 5, Issue 4, August 2016, Page: 113-119
The QR Method for Determining All Eigenvalues of Real Square Matrices
Eyaya Fekadie Anley, College of Natural Science, Department of Mathematics, Arba Minch University, Arba Minch, Ethiopia
Received: Nov. 23, 2015;       Accepted: Dec. 3, 2015;       Published: Jul. 23, 2016
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.
Keywords
QR Method, Real Matrix, Eigen Value
Eyaya Fekadie Anley, The QR Method for Determining All Eigenvalues of Real Square Matrices, Pure and Applied Mathematics Journal. Vol. 5, No. 4, 2016, pp. 113-119. doi: 10.11648/j.pamj.20160504.15
Reference

Bronson, Rchard. (1991). Matrix Methods: An introduction. 2nd ed. San Diego: Academic press, inc.

Faires, J. D and R. L. burden., (2002). Numerical Methods 3rd ed. Vol. 2. Pblisher: Broks cole

Horn, R. A. and C. A. Johnson., (1985). Matrix Analysis. 1st ed. Cambridge: Cambridge University.

H. R. saxena. (2000). Finite Difference & Numerical Analysis. Pub S. Chad company LTD

Iyenger S. R. K, jain R. K. (2009), Numerical Methods, New delhi: New age international publishers.

Kres. R. (1998), Graduate Texts In Mathematics, New York: spriner-verlag.

Michelles. S. (1990). A simple proof of convergence of the QR Algorithm for Normal matrices without shifts. IMA Ppreprint series NO 720.

Muzafar F. Hama. (2010). A Technique to Have a Convergence for the QR Algorithm.

International Journal of Algebra, Vol. 6, 2012, no. 2, 65 - 72, University of Sulaimani, College of Science Department of Mathematics, Sulaimani, Iraqhamamuzafar@yahoo.com

Paul Schmitz. (2012). The QR algorithm senior seminar, university of Minnesota Morris spring.

Watkins, Davis S, (2008). The QR algorithm Revised. SIAM Review 50. 1: 133-145. 