The representations and computations of generalized inverses $$A^{(1)}_{T,S}$$ A T , S ( 1 ) , $$A^{(1,2)}_{T,S}$$ A T , S ( 1 , 2 ) and the group inverse

In this paper, we derive novel representations of generalized inverses $$A^{(1)}_{T,S}$$ and $$A^{(1,2)}_{T,S}$$ , which are much simpler than those introduced in Ben-Israel and Greville (Generalized inverses: theory and applications. Springer, New York, 2003). When $$A^{(1,2)}_{T,S}$$ is applied to matrices of index one, a simple representation for the group inverse $$A_{g}$$ is derived. Based on these representations, we derive various algorithms for computing $$A^{(1)}_{T,S}$$ , $$A^{(1,2)}_{T,S}$$ and $$A_{g}$$ , respectively. Moreover, our methods can be achieved through Gauss–Jordan elimination and complexity analysis indicates that our method for computing the group inverse $$A_{g}$$ is more efficient than the other existing methods in the literature for a large class of problems in the computational complexity sense. Finally, numerical experiments show that our method for the group inverse $$A_{g}$$ has highest accuracy among all the existing methods in the literature and also has the lowest cost of CPU time when applied to symmetric matrices or matrices with high rank or small size matrices with low rank in practice.