Computation of Moore–Penrose generalized inverses of matrices with meromorphic function entries

Abstract

In this paper, given a field with an involutory automorphism, we introduce the notion of Moore–Penrose field by requiring that all matrices over the field have Moore–Penrose inverse. We prove that only characteristic zero fields can be Moore–Penrose, and that the field of rational functions over a Moore–Penrose field is also Moore–Penrose. In addition, for a matrix with rational functions entries with coefficients in a field K, we find sufficient conditions for the elements in K to ensure that the specialization of the Moore–Penrose inverse is the Moore–Penrose inverse of the specialization of the matrix. As a consequence, we provide a symbolic algorithm that, given a matrix whose entries are rational expression over C of finitely many meromorphic functions being invariant by the involutory automorphism, computes its Moore–Penrose inverve by replacing the functions by new variables, and hence reducing the problem to the case of matrices with complex rational function entries.