A Class of Quasi-Sparse Companion Pencils
In this paper, we introduce a general class of quasi-sparse potential companion pencils for arbitrary square matrix polynomials over an arbitrary field, which extends the class introduced in [B. Eastman, I.-J. Kim, B. L. Shader, K.N. Vander Meulen, Companion matrix patterns. Linear Algebra Appl. 436 (2014) 255–272] for monic scalar polynomials. We provide a canonical form, up to permutation, for companion pencils in this class. We also relate these companion pencils with other relevant families of companion linearizations known so far. Finally, we determine the number of different sparse companion pencils in the class, up to permutation.
KeywordsCompanion matrix Companion pencil Linearization Sparsity Scalar polynomial Matrix polynomial arbitrary field Permutation
We wish to thank two anonymous referees for the careful reading of this paper and for their comments that allowed us to improve the presentation.
This work has been partially supported by the Ministerio de Economía y Competitividad of Spain through grants MTM2015-68805-REDT and MTM2015-65798-P.