An efficient method for computing the outer inverse AT, S(2) through Gauss-Jordan elimination

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.

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