Summary: For Hermitian matrices and generalized definite eigenproblems, the $LDL^H$ factorization provides an easy tool to slice the spectrum into two disjoint intervals. In this note we generalize this method to nonlinear eigenvalue problems allowing for a minmax characterization of (some of) their real eigenvalues. In particular we apply this approach to several classes of quadratic pencils.

15A18, 65F15

eigenvalue, variational characterization, minmax principle, Sylvester's law of inertia