Weighted inner inverse for rectangular matrices

Abstract

To extend the notation of inner inverses, we define weighted inner inverses of a rectangular matrix. In particular, we introduce a W -weighted (B, C)-inner inverse of A, for given matrices A, W, B, C, and present some characterizations and conditions for its existence. Since this new inverse is not unique, we describe the set of all W -weighted (B, C)-inner inverses of a given matrix. Several invertible matrix expressions which involve a W -weighted (B, C)-inner inverse of A are studied. Using such expressions, we represent a W -weighted (B, C)-inner inverse of some other matrix E. As a particular case of W -weighted (B, C)-inner inverse, we investigate (B, C)-inner inverses of a rectangular matrix. We also establish some reverse order law properties considering weighted inner inverses.