The weak group-star matrix

In this paper, we introduce one type of matrix, called the weak group-star matrix. We investigate the characterizations, representations, and properties of the matrix. A variant of the successive matrix squaring computational iterative scheme is given for calculating the weak group-star matrix. Moreover, the Cramer's rule for the solution of a singular equation (A dagger)'x = b is presented. Then, the perturbation is also given for the weak group-star matrix. In the final, the weak group-star matrix being used in solving appropriate systems of linear equations is established.