Abstract: An important characterization formula for M{2} was given by Stewart where M 2 Cm×n. But this formula contains redundant arbitrary parameters, and therefore is nonefficient. This paper, by using the matrix full rank decomposition, showed that for a proper subset of I{2}s, which is denoted as B1, the redundant arbitrary parameters in Stewart’s formula can be eliminated, and I{2}s is a union set of its certain subsets, and each of the subsets is 2-norm isometry with B1. Finally, the efficient characterization fonmulas for I{2}s, I{2} and M{2} are obtained respectively. An algorithm was provided that can be used to compute any element of I{2}s, and avoid the repeat computation work for each
element of I{2}s.

Key words: efficient characterization for a generalized inverse matrix set, 2-norm isometry of two sets, matrix full-rank factorization, solution set of a matrix equation