Variational principles for indefinite eigenvalue problems

Abstract

Let A and B be N × N complex Hermitian matrices where B is nonsingular but neither A nor B need be definite. Let S k denote the linear subspaces of C N of codimension k, and let σ ± k = sup{ inf{x∗ Ax : x∗ Bx = ±1, x ∈ S} : S ∈ S k−1} . Assuming that the real eigenvalues λ of the problem Aχ = λBχ are semisimple, we show how to calculate the value of each σ ± k and the corresponding inf sups. In consequence we show precisely which eigenvalues of (∗) can be obtained by variational formulae of the above types.