Non-sparse Companion Matrices

Abstract

Given a polynomial $p(z)$, a companion matrix can be thought of as a simple template for placing the coefficients of $p(z)$ in a matrix such that the characteristic polynomial is $p(z)$. The Frobenius companion and the more recently-discovered Fiedler companion matrices are examples. Both the Frobenius and Fiedler companion matrices have the maximum possible number of zero entries, and in that sense are sparse. In this paper, companion matrices are explored that are not sparse. Some constructions of non-sparse companion matrices are provided, and properties that all companion matrices must exhibit are given. For example, it is shown that every companion matrix realization is non-derogatory. Bounds on the minimum number of zeros that must appear in a companion matrix, are also given.

Companion unit lower Hessenberg matrices

“…A consequence of Theorem 7 is related with an open question stated by Deaett et al [3]: "We wonder if, in producing a companion matrix by changing some zero entries of a Fiedler companion matrix F i 1 ,...,in by nonzero constants, the extra nonzero entries are always restricted to the submatrix corresponding to the i 1 -block ". They partially confirmed that supposition.…”

“…When we broaden our gaze to non-sparse companion matrices we will no longer obtain a result like Theorem 1. This is so because Deaett et al [3] showed one non-sparse companion matrix that can not be transformed into a ULH matrix by any combination of transposition, permutation similarity and diagonal similarity. So we will focus our efforts to obtain a result like Theorem 2 also valid for non-sparse companion ULH matrices.…”

“…where A is a constant ULH matrix of order 6, and B is a constant ULH matrix of order [1,2,3,4,5] and A [1,2,3,4,5,6] are nilpotent. The ULH submatrices A [1] and A [1,2] have no restriction.…”

“…In 2014 Eastmann et al [5,6] showed that every sparse companion matrix can be transformed into a ULH matrix by any combination of transposition, permutation similarity and diagonal similarity (we will be more precise in Theorems 1 and 2 below). In 2019 Deaett et al [3] gave one example that shows that this is not necessarily the case for non-sparse companion matrices. On the other hand, they proved that any companion matrix is nonderogatory.…”

A note on generalized companion pencils in the monomial basis

“…During the refereeing process of this manuscript we have been informed by K. N. Vander Meulen on the recent reference [24], that deals with companion matrices for real monic polynomials, and where it is proved ([24, Theorem 3.1]) that they are nonderogatory. However, we want to emphasize that the notion of companion matrix used in that reference imposes the restriction that each coefficient a i appears just once in the matrix, and that the remaining entries are constant.…”