Infimum of a pair of matrices

Abstract

In this paper answers are provided to the following questions: Given a matrix E1 of order k × m, does there exist a matrix E2 of order (m-k)×m such that the matrix E=(E1E2) is square, idempotent and of the same rank as E1? Further, when is the choice of E2 unique? Given a subspace T, virtually disjoint with the row span of E1, when does there exist a matrix G=(G1G2) such that G is square idempotent of rank equal to rank G1=rank E1 and G1 is congruent to E1 (mod T)? These results are then used to determine if a pair of matrices A and B has an infimum in the minus partial order. Further, the existence of the sharp order infimum is studied.