Variational principles for indefinite eigenvalue problems Original Research Article

Let A and B be N × N complex Hermitian matrices where B is nonsingular but neither A nor B need be definite. Let Sk denote the linear subspaces of imageN of codimension k, and let σ±k = sup{inf{x* Ax : x* Bx = ±1, x set membership, variant S} : S set membership, variant Sk−1}. Assuming that the real eigenvalues λ of the problem imageAχ = 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.