On the variation of the spectra of matrices

Several “distances” between the spectra of two matrices are discussed and compared. Optimal bounds are given, which enable us to reduce certain bounds on the eigenvalue variation of matrices by a factor of about two. The results of Bhatia and Mukherjea and of Bhatia and Friedland on the eigenvalue variation are derived in an elementary way using results of Henrici on the spectral variation.