On computing root polynomials and minimal bases of matrix pencils

We revisit the notion of root polynomials, thoroughly studied in (Dopico and Noferini, 2020 [9]) for general polynomial matrices, and show how they can efficiently be computed in the case of a matrix pencil lambda E+ A. The method we propose makes extensive use of the staircase algorithm, which is known to compute the left and right minimal indices of the Kronecker structure of the pencil. In addition, we show here that the staircase algorithm, applied to the expansion (lambda -lambda(0))E+(A - lambda E-0), constructs a block triangular pencil from which a minimal basis and a maximal set of root polynomials at the eigenvalue lambda(0), can be computed in an efficient manner. (c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).