On the Rodman-Shalom conjecture regarding the Jordan form of completions of partial upper triangular matrices

Abstract

Rodman and Shalom [Linear Algebra and its Applications, 168:221-249, 1992] conjecture the following statement: Let A be a lower irreducible partial upper triangular n x n matrix over F such that trace(A) = 0. Let n1 ≥ n2 ≥ ⋯ np ≥ 1 be a set of p positive integers such that ∑i=1p ni = n. There exists a nilpotent completion Ac of A whose Jordan form consists of p blocks of sizes ni x ni , i = 1,2..., p if and only if r(A k) ≤ ∑i:ni≥ k (ni-k), k = 1,2,..., n1. In this paper this conjecture is solved in two cases: when the minimal rank of A is 2, and for matrices of size 5 x 5.