A note on the backward errors for Hermite eigenvalue problems

We discuss the backward error analysis of the Hermite eigenproblems. Three kinds of problems are included: generalized eigenproblem, matrix polynomial eigenproblem, and multiparameter eigenproblem. In the case of the approximate eigenvalue being real, these problems have already been solved by Higham and Higham [SIAM J. Matrix Anal. 20 (1998) 493], Tisseur [Lin. Alg. Appl. 309 (2002) 339], and Hachstenbach and Plestenjak [Lin. Alg. Appl. 375 (2003) 63]. This note as a complement of their results considers the complex case.