Periodic matrix difference equations and companion matrices in blocks: some applications

This study is devoted to some periodic matrix difference equations, through their associated product of companion matrices in blocks. Linear recursive sequences in the algebra of square matrices in blocks and the generalized Cayley–Hamilton theorem are considered for working out some results about the powers of matrices in blocks. Two algorithms for computing the finite product of periodic companion matrices in blocks are built. Illustrative examples and applications are considered to demonstrate the effectiveness of our approach.

Nonsparse companion Hessenberg matrices

In recent years, there has been a growing interest in companion matrices. Sparse companion matrices are well known: every sparse companion matrix is equivalent to a Hessenberg matrix of a particular simple type. Recently, Deaett et al. [Electron. J. Linear Algebra, 35:223--247, 2019] started the systematic study of nonsparse companion matrices. They proved that every nonsparse companion matrix is nonderogatory, although not necessarily equivalent to a Hessenberg matrix. In this paper, the nonsparse companion matrices which are unit Hessenberg are described. In a companion matrix, the variables are the coordinates of the characteristic polynomial with respect to the monomial basis. A PB-companion matrix is a generalization, in the sense that the variables are the coordinates of the characteristic polynomial with respect to a general polynomial basis. The literature provides examples with Newton basis, Chebyshev basis, and other general orthogonal bases. Here, the PB-companion matrices which are unit Hessenberg are also described.

Bounds for the zeros of polynomials from compression matrix inequalities

In this paper, some compression matrix inequalities are applied to the Frobenius companion matrices of monic polynomials in order to obtain new upper bounds for the zeros of such polynomials.

m-NIL-CLEAN COMPANION MATRICES

Companion matrices over fields of prime characteristic, p, that are sums of two idempotents and a nilpotent are characterized in terms of dimension and trace of such a matrix and of p. Companion matrices over fields of positive characteristic, p, that are sums of m idempotents, m ≥ 2, and a nilpotent are characterized in terms of dimension and trace of such a matrix and of p.