The Modified Power Method for Solving the Eigenvalue Problem with the Use of Idempotent Matrix

Abstract:

The work presents a new approach to the power method serving the purpose of solving the eigenvalue problem of a matrix. Instead of calculating the eigenvector corresponding to the dominant eigenvalue from the formula , the idempotent matrix B associated with the given matrix A is calculated from the formula , where m stands for the method’s rate of convergence. The scaling coefficient k_i is determined by the quotient of any norms of matrices B_i and or by the reciprocal of the Frobenius norm of matrix B_i. In the presented approach the condition for completing calculations has the form. Once the calculations are completed, the columns of matrix B are vectors parallel to the eigenvector corresponding to the dominant eigenvalue, which is calculated from the Rayleigh quotient. The new approach eliminates the necessity to use a starting vector, increases the rate of convergence and shortens the calculation time when compared to the classic method.

• Periodical: Applied Mechanics and Materials (Volume 712)
• Pages: 43-48
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.712.43
• Online since: January 2015
• Authors: Rafał Palej*, Artur Krowiak, Renata Filipowska
• Keywords: Eigenvalue Problem, Idempotent Matrix, Power Method, Rate of Convergence

References

[1] I. Ipsen, R.M. Wills, Analysis and Computation of Google's PageRank, 7th IMACS International Symposium on Iterative Methods in Scientific Computing, Fields Institute, Toronto, Canada, (2005).

[2] P. Gupta, A. Goel, J. Lin, A. Sharma, D. Wang, R.B. Zadeh, WTF: The who-to-follow system at Twitter, Proceeding of the 22nd International conference on World Wide Web, Geneva, (2013).

DOI: 10.1145/2488388.2488433

[3] D. S. Bernstein, Matrix Mathematics, Princeton University Press, New Jersey, (2005).

[4] R. Palej, Matrix eigenvalue problem in the theory of vibrations, Cracow Univ. of Technology, Kraków, 2010 [in Polish].