New results on algorithms for the computation of output-nulling and input-containing subspaces

In this paper we present and provide a proof for a set of non-recursive formulae arising in the computation of the largest output-nulling and the smallest input-containing subspaces which have been used in a variety of contexts in the framework of the geometric approach. These expressions have been used in the literature both in the strictly proper and in the non-strictly proper case, but, to the best of our knowledge, a proof is still missing. These formulae are established here in the general possibly non-strictly proper case. Some ancillary side results of independent interest are also proposed.