A volume associated with generalized inverse A_(T,S)~(2)

The volume of a matrix A ∈ Rrm×n,denoted by vol (A) ,is defined as the product of its nonzero singular values. So the volume of the Moore-Penrose inverse A+ is given directly by vol(A+)=1/vol(A).It means that the expression of the volume of the Moore-Penrose inverse A+ can be obtained without calculating the Moore-Penrose inverse A+ firstly. This paper deals with the representation of the volume of the generalized inverse AT,S(2) without calculating the generalized inverse AT,S(2) firstly. The expressions of the volumes of other generalized inverses, includeing the weighted M-P inverse, the group inverse,and the Drazin inverse,are provided. The main results are also the generalization of the results proposed by [3].