On perturbations of matrix pencils with real spectra. II

Abstract:

A well-known result on spectral variation of a Hermitian matrix due to Mirsky is the following: Let $A$ and $\widetilde A$ be two $n\times n$ Hermitian matrices, and let $\lambda _1$, …, $\lambda _{n}$ and $\widetilde \lambda _1$, …, $\widetilde \lambda _{n}$ be their eigenvalues arranged in ascending order. Then $\left \Vvert \operatorname {diag} (\lambda _1- \widetilde \lambda _1,\ldots ,\lambda _n- \widetilde \lambda _n) \right \Vvert \le \left \Vvert A-\widetilde A \right \Vvert$ for any unitarily invariant norm $\Vvert \cdot \Vvert$. In this paper, we generalize this to the perturbation theory for diagonalizable matrix pencils with real spectra. The much studied case of definite pencils is included in this.