Maximal orthogonality and pseudo-orthogonality with applications to generalized inverses Original Research Article

Equip a finite-dimensional vector space V over a field F with a nondegenerate symmetric bilinear, alternating bilinear, or hermitian form. Fix some subspace U of V. The concept of a maximally orthogonal complementary subspace for U is presented and shown to be related to the concept of a pseudo-orthogonal complementary subspace from an earlier paper. When F = GF(q), the number of maximally orthogonal (pseudo-orthogonal) complements is computed. Also, both types of complements are characterized as orthogonal complements with respect to related forms. This is used to relate certain reflexive generalized inverses of linear transformations to Moore–Penrose inverses. Techniques are presented for deriving the related forms, which, together with standard techniques for computing Moore–Penrose inverses, can be used to compute the desired generalized inverses. Lastly, vectors of V that occur in such complementary subspaces of U are characterized.