A Closed Formula for the Moore-Penrose Generalized Inverse of a Complex Matrix of Given Rank

For the Moore-Penrose generalized inverse of complex matrices, we establish closed forms valid on each set of matrices of given rank. These expressions are matrices of rational functions of the matrix coefficients and their complex conjugates, so that it is seen explicitly and constructively that taking the Moore-Penrose inverse is a real-analytic operation when restricted to the subsets of matrices of given rank.