Generalized inverse of linear transformations: a geometric approach

Abstract

A generalized inverse of a linear transformation A: V→W, where V and W are arbitrary finite dimensional vector spaces, is defined using only geometrical concepts of linear transformations. The inverse is uniquely defined in terms of specified subspaces ℒ⊂W, M⊂V and a linear transformation N satisfying some conditions. Such an inverse is called the ℒMN-inverse. A Moore-Penrose type inverse is obtained by choosing N = 0. Some optimization problems are considered by choosing V and W as inner product spaces. Our results extend without any major modification of proofs to bounded linear operators with closed range on Hilbert spaces.