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Journal of Applied Mathematics and Computing

, Volume 50, Issue 1–2, pp 389–416 | Cite as

Spectral properties of the matrix \(C^{-1} B\) with positive definite \(C\) and Hermitian \(B\) as well as applications

  • L. KohauptEmail author
Original Research

Abstract

In this paper, spectral properties of the matrix \(C^{-1} B\) are derived where \(C\) is positive definite and \(B\) is Hermitian. Two special cases are considered. In the first case, \(C=R\) and \(B=\frac{A^{*}R+R A}{2}\) (resp. \(B=\frac{R A-A^{*}R}{2 i}\)) where \(A\) is a diagonalizable matrix and \(R\) is a positive definite matrix constructed from the eigenvectors of \(A^{*}\). It will be shown that the eigenvalues of \(R^{-1}\frac{A^{*}R+R A}{2}\) are equal to the real parts (resp. that the eigenvalues of \(R^{-1}\frac{R A-A^{*}R}{2 i}\) are the imaginary parts) of the eigenvalues of \(A\) and that the eigenvectors of \(R^{-1}\frac{A^{*}R+R A}{2}\) (resp. of \(R^{-1} \frac{R A-A^{*}R}{2 i}\)) are equal to those of \(A\), which is a new result and of interest on its own. This leads to two-sided estimates on the real parts (resp. on the imaginary parts) of the eigenvalues of matrix \(A\) that improve existing ones. Numerical examples underpin the theoretical findings. In the second case, \(C=R\) and \(B=A^{*}R\, A\) where \(A\) is a diagonalizable matrix. It will be shown that the eigenvalues of \(R^{-1}A^{*}R\, A\) are equal to the squared moduli of the eigenvalues of \(A\) and that the eigenvectors of \(R^{-1}A^{*}R\, A\) are equal to those of the eigenvectors of \(A\), which is also a new result and of interest on its own.

Keywords

Real and imaginary parts of eigenvalues Squared moduli of eigenvalues Two-sided estimates Weighted norm  Dynamical system State-space description 

Notes

Acknowledgments

The author would like to give thanks to anonymous referees for their comments that led to a better presentation of the paper.

References

  1. 1.
    Czornik, A., Jurgaś, P.: Some properties of the spectral radius of a set of matrices. Int. J. Appl. Math. Sci. 16(2), 183–188 (2006)zbMATHGoogle Scholar
  2. 2.
    Kohaupt, L.: Construction of a biorthogonal system of principal vectors of the matrices \(A\) and \(A^{\ast }\) with applications to the initial value problem \(\dot{x}=A\, x, \; x(t_0)=x_0\). J. Comput. Math. Optim. 3(3), 163–192 (2007)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Kohaupt, L.: Biorthogonalization of the principal vectors for the matrices \(A\) and \(A^{\ast }\) with application to the computation of the explicit representation of the solution \(x(t)\) of \(\dot{x}=A\, x, \; x(t_0)=x_0\). Appl. Math. Sci. 2(20), 961–974 (2008)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Kohaupt, L.: Solution of the matrix eigenvalue problem \(V A + A^{\ast } V = \mu V\) with applications to the study of free linear systems. J. Comput. Appl. Math. 213(1), 142–165 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kohaupt, L.: Solution of the vibration problem \(M\ddot{y}+B\dot{y}+K y = 0, \, y(t_0)=y_0, \, \dot{y}(t_0)=\dot{y}_0\) without the hypothesis \(B M^{-1} K = K M^{-1} B\) or \(B = \alpha M + \beta K\). Appl. Math. Sci. 2(41), 1989–2024 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Kohaupt, L.: On the vibration-suppression property and monotonicity behavior of a special weighted norm for dynamical systems \(\dot{x}=A x, \, x(t_0)=x_0\). Appl. Math. Comput. 222, 307–330 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Lancaster, P.: Theory of Matrices. Academic Press, New York (1969)zbMATHGoogle Scholar
  8. 8.
    Laffey, T.J., S̆migoc, H.: Nonnegatively realizable spectra with two positive eigenvalues. Linear Multilinear Algebra 58(7–8), 1053–1069 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Savchenko, S.V.: On the change in the spectral properties of a matrix under perturbations of sufficiently low rank. Funct. Anal. Appl. 38(1), 69–71 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, 3rd edn. Springer, New York (2010)Google Scholar
  11. 11.
    Stummel, F., Hainer, K.: Introduction to Numerical Analysis. Scottish Academic Press, Edinburgh (1980)zbMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2015

Authors and Affiliations

  1. 1.Department of MathematicsBeuth University of Technology BerlinBerlinGermany

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