We define an idempotent-aided factorization of a matrix D, with the help of an idempotent matrix E of the same rank as D, as the representation of D in the form \(D=UV\) , where the matrix U has the same range as D and the same null space as E, while the matrix V has the same null space as D and the same range as E. Such factorizations can be considered as a natural generalization of full rank factorizations. We provide three efficient algorithms for determining idempotent-aided factorizations of matrices over a field, as well as the fourth one that determines the canonical idempotent-aided factorization. We also apply those algorithms in the construction of algorithms for testing the existence and computing group inverses and (B, C)-inverses of matrices over a field.
