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In this paper, two novel closed formulas for the generalized inverseAT,S(2){A_{T,S}^{(2)}}of a complex matrix  A  are established based on the expressionAT,S(2)=G⁢(A⁢G)g=(G⁢A)g⁢G{A_{T,S}^{(2)}=G(AG)_{g}=(GA)_{g}G}, according to the matrix  G  with given different ranks, where  G  is a matrix such thatR⁢(G)=T{R(G)=T}andN⁢(G)=S{N(G)=S}. The two formula express each component ofAT,S(2){A_{T,S}^{(2)}}as a rational function of the components of the matrices  A  and  G , the efficiency of which is shown by some numerical examples.

journal of inverse and ill-posed problems

Two closed novel formulas for the generalized inverse  A                                      T , S                                   (2)  of a complex matrix with given rank