Variational principles for indefinite eigenvalue problems
The history of the development of variational principles for the selfadjoint generalized eigenvalue problem \(Ax = \lambda Bx\) \((x \neq 0)\) is retrospected in some detail firstly. Restricting \(A\) and \(B\) to be \(n \times n\) complex Hermitian matrices with \(B\) nonsingular but neither \(A\) nor \(B\) definite, the authors generalize all the cited results for the matrix case with semisimple real eigenvalues, allowing nonreal or non- semisimple ones. Let \(S_ k\) denote the linear subspaces of \(\mathbb{C}^ N\) of codimension \(k\) and \(\sigma^ \pm_ k = \text{sup} \{\inf \{x^* Ax : x^* Bx = \pm 1,\;x \in S\} : S \in S_{k - 1}\}\). They show how to calculate the value of each \(\sigma^ 2_ k\) and the corresponding inf sups and in consequence, indicate precisely which eigenvalues of \(Ax = \lambda Bx\) can be obtained by variational formulae of the above types. A simple algorithm is also given for testing exactly which eigenvalues (including those of the mixed type) of this equation can be characterized by \(\sigma^ \pm_ k\) or \(\delta^ \pm_ k\), some of these characterizations being new. This algorithm is illustrated on an example, which shows some of the differences between the cited results.
selfadjoint generalized eigenvalue problem
scientific article; zbMATH DE number 753873
scientific article