Bor Plestenjak: Singular eigenvalue problems

We say that a generalized eigenvalue problem is singular if its characteristic polynomial is identically zero. In this case, eigenvalues are defined as points where the rank of the pencil drops. Similar notions arise for singular polynomial and multiparameter eigenvalue problems.

Singular eigenvalue problems are highly challenging, both from the viewpoints of theory and numerical computation: even the proper definition of eigenvectors may become nontrivial, while achieving accuracy and computational efficiency remains difficult.

We present a simple method for generalized eigenvalue problems based on a rank-completing perturbation of the original problem, together with several open questions and applications leading to singular eigenvalue problems in various forms.