A Class of Quasi-Sparse Companion Pencils

Abstract

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.

A note on generalized companion pencils in the monomial basis

“…In [19], we looked for companion pencils (as in Definition 21) with a small number of nonzero entries. There, however, only nonzero entries of the form a i , λa i+1 or λa i+1 + a i (up to scalar constants) were considered, and this led to the class of quasi-sparse pencils with at most 3k − 2 nonzero entries, denoted by R n,k (see [19,Def. 2…”

“…This is, precisely, the notion of generalized companion pencil we introduce in Definition 22, which is an extension, to matrix pencils, of the notion of generalized companion matrix in [30]. We use the term "generalized" because it is the one used in [30], and also in order to be consistent with previous work of the authors [19,Def. 2.2], where companion pencil is used for a more restrictive notion.…”

“…In [19,26,35], special attention has been paid to the notion of sparsity, namely the property of having the smallest possible number of nonzero entries. In particular, it has been proved in [35] that the smallest number of nonzero entries in a companion matrix for scalar polynomials of degree k is 2k−1, and, in [26], a canonical expression (up to permutation similarity) has been presented for the so-called sparse companion patterns.…”

“…In particular, it has been proved in [35] that the smallest number of nonzero entries in a companion matrix for scalar polynomials of degree k is 2k−1, and, in [26], a canonical expression (up to permutation similarity) has been presented for the so-called sparse companion patterns. Up to our knowledge, the only reference containing a study on similar issues for companion pencils is the recent work [19]. In that reference, a canonical expression (up to permutation equivalence) is also presented for a general class of so-called quasi-sparse companion pencils, which resembles the class of companion patterns in [26].…”

“…The present work is a complement to [19]. In particular, we deal not only with companion pencils (including those in the family introduced in [19]), but also with generalized companion pencils (see Definitions 21 and 22), where we allow the non-constant entries of the pencil to be not just the coefficients, a i , of the polynomial (1), but polynomials in these coefficients. We are also interested in the sparsity of these constructions, namely in determining the smallest possible number of nonzero entries (these issues are addressed in Section 5).…”