Partition, Construction, and Enumeration of M–P Invertible Matrices over Finite Fields

A necessary and sufficient condition for an m×n matrix A over Fq having a Moor–Penrose generalized inverse (M–P inverse for short) was given in (C. K. Wu and E. Dawson, 1998, Finite Fields Appl. 4, 307–315). In the present paper further necessary and sufficient conditions are obtained, which make clear the set of m×n matrices over Fq having an M–P inverse and reduce the problem of constructing M–P invertible matrices to that of constructing subspaces of certain type with respect to some classical groups. Moreover, an explicit formula for the M–P inverse of a matrix which is M–P invertible is also given. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing results in geometry of classical groups over finite fields (Z. X. Wan, 1993, “Geometry of Classical Groups over Finite Fields”, Studentlitteratur, Chatwell Bratt).