Computation of generalized inverses by using the LDL∗LDL∗ decomposition

An efficient algorithm, based on the LDL∗LDL∗ factorization, for computing {1,2,3}{1,2,3} and {1,2,4}{1,2,4} inverses and the Moore–Penrose inverse of a given rational matrix AA, is developed. We consider matrix products A∗AA∗A and AA∗AA∗ and corresponding LDL∗LDL∗ factorizations in order to compute the generalized inverse of AA. By considering the matrix products (R∗A)†R∗(R∗A)†R∗ and T∗(AT∗)†T∗(AT∗)†, where RR and TT are arbitrary rational matrices with appropriate dimensions and ranks, we characterize classes A{1,2,3}A{1,2,3} and A{1,2,4}A{1,2,4}. Some evaluation times for our algorithm are compared with corresponding times for several known algorithms for computing the Moore–Penrose inverse.