On perturbations of matrix pencils with real spectra. II

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 $% \lambda_1,\ldots,\lambda_{n}$ and $% \widetilde\lambda_1,\ldots,\widetilde\lambda_{n}$ be their eigenvalues arranged in ascending order. Then $% \left|\kern-1.5truept\left|\kern-1.5truept\left| \operatorname{diag}\, (\lambda_1-% \widetilde\lambda_1,\ldots,\lambda_n-% \widetilde\lambda_n)% \right|\kern-1.5truept\right|\kern-1.5truept\right| \le% \left|\kern-1.5truept\left|\kern-1.5truept\left| A-\widetilde A% \right|\kern-1.5truept\right|\kern-1.5truept\right|$ for any unitarily invariant norm $% \hbox{$|\kern-1.5truept|\kern-1.5truept|\cdot |\kern-1.5truept|\kern-1.5truept|$}$. 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.