An Efficient Factorization for the Group Inverse

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An efficient algorithm is introduced for computing the group inverse of a square, singular matrix, in factorized form. The algorithm is based on the QR factorization with column pivoting and uses a technique of inversion by partitioning. The factorization is used to compute the group inverse solution of a singular system of equations. When only the solution vector is wanted, the group inverse does not need to be computed explicitly.