Automatic recovery of eigenvectors and minimal bases of matrix polynomials from generalized Fiedler pencils with repetition

Linearization is a widely used method for solving polynomial eigenproblems in which a matrix polynomial is transformed to a matrix pencil of larger size. Fiedler pencils, generalized Fiedler pencils, Fiedler pencils with repetition (FPRs) and generalized Fiedler pencils with repetition (GFPRs) are well known classes of strong linearizations of matrix polynomials. The class GFPRs is also a rich source of structure-preserving strong linearizations of structured matrix polynomials. The recovery of eigenvectors, minimal bases and minimal indices of matrix polynomials from those of the linearizations is an important task. It is well known that eigenvectors and minimal bases of matrix polynomials can be easily recovered from those of the Fiedler and generalized Fiedler pencils. By contrast, barring a small subclass of FPRs, the recovery of eigenvectors and minimal bases of matrix polynomials from those of the FPRs and GFPRs is still an open problem. The purpose of this paper is to fill this gap. We consider the class of GFPRs, which subsumes the class of FPRs, and describe the recovery of eigenvectors, minimal bases and minimal indices of matrix polynomials from those of the GFPRs and show that the recovery is operation-free. (C) 2019 Elsevier Inc. All rights reserved.