Journal of Applied Mathematics and Computing

, Volume 50, Issue 1–2, pp 389–416

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

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

## 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)
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)
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)
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)
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)
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)
7. 7.
Lancaster, P.: Theory of Matrices. Academic Press, New York (1969)
8. 8.
Laffey, T.J., S̆migoc, H.: Nonnegatively realizable spectra with two positive eigenvalues. Linear Multilinear Algebra 58(7–8), 1053–1069 (2010)
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)
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) 