Further spectral properties of the matrix $C^{-1} B$ with positive definite C and Hermitian B applied to wider classes of matrices C and B

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 general square matrix and R is a positive definite matrix constructed from the principal vectors of $A^{*}.$ It will be shown that the eigenvalues of $R^{-1}\frac{A^{*}R+R A}{2}$ are only then equal to the real parts (resp. that the eigenvalues of $R^{-1}\frac{R A-A^{*}R}{2 i}$ are only then the imaginary parts) of the eigenvalues of A if the algebraic multiplicities of the pertinent eigenvalues of A are equal to unity. In this case, 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 new two-sided estimates on the real parts (resp. on the imaginary parts) of the eigenvalues of matrix A. A numerical example underpins the theoretical findings. In the second case, $C=R$ and $B=A^{*}R A$ where A is again a general square matrix. It will be shown that the eigenvalues of $R^{-1}A^{*}R A$ are only then equal to the squared moduli of the eigenvalues of A if the algebraic multiplicity of these eigenvalues is equal to unity, and that, in this case, the pertinent eigenvectors of $R^{-1}A^{*}R A$ are equal to those of the corresponding eigenvectors of A,  which is also a new result and of interest on its own. This paper generalizes results of an earlier paper where matrix A was assumed to be diagonalizable.