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Dedicated to Hans Schneider on the occasion of his seventieth birthday.jecture the following statement: Let A be a lower irreducible partial upper triangular n n matrix over F such that traceA = 0. L e t n 1 n 2 np 1 be a set of p positive i n tegers such that p X i=1 n i = n. There exists a nilpotent completion Ac of A whose Jordan form consists of p blocks of In this paper this conjecture is solved in two cases: when the minimal rank of A is 2, and for matrices of size 5 5.

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