The strongly Bott-Duffin (E, F)-inverse for rectangular matrices

Motivated by the recently introduced strongly Bott-Duffin (e,f)-inverse for elements in a semi-group [M. Drazin, Linear Multilinear Algebra, 71(8) (2023), 1397-1406], the aim of this paper is to investigate this notion in the context of complex rectangular matrices. We provide necessary and sufficient conditions for its existence and derive a general expression involving an arbitrary inner inverse of the matrix. In addition, we obtain a canonical form for this inverse via the classical singular value decomposition. Furthermore, we show that the recently introduced generalized bilateral inverse, as well as several other notions appearing in the recent literature, can be viewed as particular cases of this new inverse.