Totally isotropic subspaces, complementary subspaces, and generalized inverses

Abstract

Let us fix a field F, a finite-dimensional F-vector space V, and a nondegenerate symmetric bilinear form on V, subject to the following restriction. If char( F) = 2, then the bilinear form must be selected so that the space of all isotropic vectors in V is nondegenerate. Let N be the set of all totally isotropic subspaces of V. There exists a mapping p: N → N( U → U p such that U + U p is nondegenerate for all U ϵ N. From such, a construction is given for obtaining a “pseudoorthogonal” complementary subspace for any subspace of V. Based on this construction, it is shown how to construct generalized inverses of linear transformations on V whose associated projection maps are normal linear transformations. The resulting operation for obtaining a generalized inverse has the additional property that it commutes with the operation of taking adjoints. When char( F) ≠ 2, it is shown that p can be selected so as to be an involution. For this case, constructions of such p are presented. The constructions which are derived from these, as outlined above, are then also involutory. Moreover, when F is an ordered field, p may be selected so as to be an involutory automorphism of the partially ordered set ( N, ⊆).