A note on an open question on ω- and τ- Matrices

AbstractIn a recent paper by Engel and Schneider, it was asked if, for every n ⩾ 1, A ∈ τ<n> implies (A+D) ∈ τ<n> for every D = diag[d1, d2,… dn] with di ⩾ 0, 1 ⩽ i ⩽ n. We answer this question in the negative. More precisely, we show that for, any n ⩾ 3, the set (τ < n>): = {D ∈Cn,n:(A+D)∈τ < n> for all A∈τ<n>} is exactly given by(Gt<n>) = {γIn:γ ⩾ 0}