Advertisement

Collinearity Models in the Eigenvalue Problem

  • Leon Bobrowski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10751)

Abstract

Solution of the eigenvalue problems can be based on inverting matrices built from regularized vectors. The regularization parameters are equal to the eigenvalues of the given matrix after fitting in accordance to the collinearity models. In this approach the eigenvectors are equal to the columns of the inverted matrix.

A new procedure of matrix inversion with the basis exchange algorithm has been recently proposed. In accordance with this procedure the inverse matrix is computed in an iterative manner by gradual replacement of unit vectors by successive regularized vectors. Such replacement is impossible if a new regularized vector depends linearly on the vectors which are already in the basis.

Keywords

Eigenvalue problem Iterative inversion of matrices Fitting eigenvalues Collinearity models 

Notes

Acknowledgments

The presented study was supported by the grant S/WI/2/2017 from Bialystok University of Technology and funded from the resources for research by Polish Ministry of Science and Higher Education.

References

  1. 1.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, 4th edn. Johns Hopkins University Press, Baltimore (2013)zbMATHGoogle Scholar
  2. 2.
    Duda, O.R., Hart, P.E., Stork, D.G.: Pattern Classification. Wiley, New York (2001)zbMATHGoogle Scholar
  3. 3.
    Jolliffe, I.T.: Principal Component Analysis. Springer, New York (2002)zbMATHGoogle Scholar
  4. 4.
    Cullum, J., Willoughby, R.A.: Large Scale Eigenvalue Problems. North-Holland (1986)Google Scholar
  5. 5.
    Saad, Y.: Numerical Methods for Large Eigenvalue Problems. Society for Industrial and Applied Mathematics, Philadelphia (2011)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bobrowski, L.: Eigenvalue problem with the basis exchange algorithm. J. Adv. Math. Comput. Sci. 23(6), 1–12 (2017). (http://www.sciencedomain.org/issue/2877)
  7. 7.
    Bobrowski, L: Large matrices inversion using the basis exchange algorithm. Br. J. Math. Comput. Sci. 21(1), 1–11 (2017). (http://www.sciencedomain.org/abstract/18203)
  8. 8.
    Bobrowski, L.: Data mining based on convex and piecewise linear criterion functions, Technical University Białystok (2005). (in Polish)Google Scholar
  9. 9.
    Bobrowski, L.: Biclustering based on collinear patterns. In: Rojas, I., Ortuño, F. (eds.) IWBBIO 2017. LNCS, vol. 10208, pp. 134–144. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-56148-6_11 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Computer ScienceBiałystok University of TechnologyBiałystokPoland

Personalised recommendations