Roth's removal rule revisited

AbstractThe theory of companion matrices is used to give explicit representations for the matrices needed in Roth's removal rule. These are then used to give simple proofs for the cyclic decomposition theorem, as well as for Roth's similarity theorem for matrices over a field

A note on light matrices

AbstractSome properties of light matrices are derived, and their relation to Perron matrices is investigated

Drazin inverses and canonical forms in Mn(Zh)

AbstractThe existence and construction of the Drazin inverse of a square matrix over the ring Zh is considered. The canonical forms of matrices over this ring are used to facilitate the computation of this type of generalized inverse

Two by two units

In this paper, we will use outer inverses and the Brown-McCoy shift to characterize the existence of the inverse and group inverse of a $2\times 2$ block matrix.FEDER Funds through ``Programa Operacional Factores de Competitividade - COMPETE'' and by Portuguese Funds through FCT - ``Fundação para a Ciência e a Tecnologia'', within the project PEst-C/MAT/UI0013/2011

Some regular sums

In this paper, we examine the question of regularity of sums of special elements that appear in the study of orthogonality and invertibility.FEDER Funds through "Programa Operacional Factores de Competitividade - COMPETE'' and by Portuguese Funds through FCT - "Fundação para a Ciência e a Tecnologia'', within the project PEst-C/MAT/UI0013/201

The (2,2,0) group inverse problem

We characterize the existence of the group inverse of a two by two matrix with zero (2,2) entry, over a ring by means of the existence of the inverse of a suitable function of the other three entries. Some special cases are derived.Fundação para a Ciência e a Tecnologia (FCT)Research Centre of Mathematics of the University of Minho (CMAT

On Roth's pseudo equivalence over rings

The pseudo-equivalence of a block lower triangular matrix T = [T_{ij}] over a regular ring and its block diagonal matrix D(T) = [T_{ii}] is characterized in terms of suitable Roth consistency conditions. The latter can in turn be expressed in terms of the solvability of certain matrix equations of the form T_{ii}X - YT_{jj} = U_{ij}.Fundação para a Ciência e a Tecnologia (FCT

Bounds on the exponent of primitivity which depend on the spectrum and the minimal polynomial

AbstractSuppose A is an n × n nonnegative primitive matrix whose minimal polynomial has degree m. We conjecture that the well-known bound on the exponent of primitivity (n − 1)2 + 1, due to Wielandt, can be replaced by (m − 1)2 + 1. The only case for which we cannot prove the conjecture is when m ⩾ 5, the number of distinct eigenvalues of A is m − 1 or m, and the directed graph of A has no circuits of length shorter than m − 1, but at least one of its vertices lies on a circuit of length not shorter than m. We show that m(m − 1) is always a bound on the exponent, this being an improvement on Wielandt's bound when m < n. For the case in which A is also symmetric, the bound which we obtain is 2(m − 1). To obtain our results we prove a lemma which shows that for a (general) nonnegative matrix, the number of its distinct eigenvalues is an upper bound on the length of the shortest circuit in its directed graph

Divisibility of finite geometric series

We give necessary and sufficient conditions for the divisibility of two finite geometric series $G_n(x)=1+x+x^2+···+x^{n−1}$ over a field of characteristic zero.This research was partially financed by Portuguese Funds through FCT (Fundação para a Ciência e a Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020. The authors thank an anonymous referee for his/her careful reading of the manuscript and valuable corrections

From euclid to corner sums - a trail of telescoping tricks

Euclid's algorithm is extended to binomials, geometric sums and corner sums. Two-sided non-commuting, non-constant linear difference equations will be solved, and the solution is applied to corner sums, thereby presenting an explicit formula for the generator of the bi-module spanned by the two starting corner sums.UIDB/00013/2020UIDP/00013/202