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

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## 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.

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