On doubly structured matrices and pencils that arise in linear response theory

Abstract

We discuss matrix pencils with a double symmetry structure that arise in the Hartree-Fock model in quantum chemistry. We derive anti-triangular condensed forms from which the eigenvalues can be read off. Ideally these would be condensed forms under unitary equivalence transformations that can be turned into stable (structure preserving) numerical methods. For the pencils under consideration this is a difficult task and not always possible. We present necessary and sufficient conditions when this is possible. If this is not possible then we show how we can include other transformations that allow to reduce the pencil to an almost anti-triangular form.