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.