Matrix pencils—theory, applications, numerical methods

This paper surveys nearly all of the publications that have appeared in the last twenty years on the theory of and numerical methods for linear pencils. The survey is divided into the following sections: theory of canonical forms for symmetric and Hermitian pencils and the associated problem of simultaneous reduction of pairs of quadratic forms to canonical form; results on perturbation of characteristic values and deflating subspaces; numerical methods. The survey is self-contained in the sense that it includes the necessary information from the elementary theory of pencils and the theory of perturbations for the common algebraic problem $Ax=\lambda x$.