Intersections and Sums of Subspaces

Abstract

The orthogonality of a pair of subspaces was defined and discussed in Chapter 12. There are pairs of subspaces that are not orthogonal but that have a weaker property called essential disjointness—orthogonal subspaces are essentially disjoint, but essentially disjoint subspaces are not necessarily orthogonal. Unlike orthogonality, essential disjointness does not depend on the choice of inner product. The essential disjointness of a pair of subspaces is defined and discussed in the present chapter. The concept of essential disjointness arises in a very natural and fundamental way in results (like those of Sections 17.2, 17.3, and 17.5) on the ranks, row and column spaces, and generalized inverses of partitioned matrices and of sums and products of matrices.