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 A and A ~ \widetilde A be two n × n n\times n Hermitian matrices, and let λ 1 \lambda _1 , …, λ n \lambda _{n} and λ ~ 1 \widetilde \lambda _1 , …, λ ~ n \widetilde \lambda _{n} be their eigenvalues arranged in ascending order. Then ⦀ diag ⁡ ( λ 1 − λ ~ 1 , … , λ n − λ ~ n ) ⦀ ≤ ⦀ A − A ~ ⦀ \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.