An efficient method for computing the outer inverse $A_{T,S}^{(2)}$ through Gauss-Jordan elimination

Abstract

In this paper, we derive a novel expression for the computation of the outer inverse $A_{T,S}^{(2)}$ . Based on this expression, we present a new Gauss-Jordan elimination method for computing $A_{R(G),N(G)}^{(2)}$ . The analysis of computational complexity indicates that our algorithm is more efficient than the existing Gauss-Jordan elimination algorithms for $A_{R(G),N(G)}^{(2)}$ in the literature for a large class of problems. Especially for the case when G is a Hermitian matrix, our algorithm has the lowest computational complexity among the existing algorithms. Finally, numerical experiments show that our method for the outer inverse $A_{R(G),N(G)}^{(2)}$ generally is more efficient than that of the other existing methods in the cases of matrices A with m < n or square matrices G with high rank or Hermitian matrices G in practice.