Nilpotent completions of partial upper triangular matrices (Q1886536)

The paper deals with different completion problems with partial upper triangular matrices. Specifically, the author presents and solves the following questions: (a) Given a partial upper triangular matrix \(A\), lower irreducible and with trace equal to zero, does there exist a nilpotent completion \(A_c\) of \(A\), such that its rank is equal to the minimal rank of \(A\)? (b) Let \(r_N\) be the minimun of the rank of all nilpotent completions of a partial matrix \(A\). If \(A\) is a partial upper triangular matrix, lower irreducible and with trace equal to zero, what is the value of \(r_N\)? (c) Let \(A\) be a partial upper triangular matrix, lower irreducible and with trace equal to zero. If \(r(A)\) denote the minimal rank of \(A\), are the following inequalities satisfied: \[ r(A^k)-r(A^{k+1}) \geq r(A^{k+1})- r(A^{k+2}), \forall k \geq 1 ? \] (d) Let \(A\) be a partial upper triangular matrix, lower irreducible and with trace equal to zero. Then, does there exist a nilpotent completion of \(A\) such that its Segre characteristic is majorized by the Segre characteristic of all possible nilpotent completion of \(A\)?