Abstract: In this paper, by {\it selecting inverse image index method}, an efficient characterization formula from set ${\mathbb{A}}_{1}=\{\alpha:\alpha=(I_s,\eta^{\rm T})^{\rm T}\in \Bbb C^{n\times s}_s\}$ onto set ${\mathbb{B}}_1=\{\beta:\beta=\alpha(\alpha^*\alpha)^{-1}\alpha^*, \alpha\in{\mathbb{A}}_1\}$ is given. Besides, it is shown that each element of $I\{2,3\}_s$ is permutation similar to an element of ${\mathbb{B}}_1$. Then efficient characterization formulas for $I\{2,3\}$ and $M\{2,3\}$ are obtained respectively. An interesting thing is ${\mathbb{B}}_1$ is a dense subset of $I\{2,3\}_s$. The fact that $I\{2,3\}_s=I\{2,4\}_s=I\{2,3,4\}_s$ enables us to obtain the efficient characterization formulas for $M\{2,4\}$ and $M\{2,3,4\}$ fluently. Algorithm 3.1 may be used to compute elements of $I\{2,3\}_s$ and to avoid the repeated computation work.

Key words: efficient characterization of classes of generalized inverses, singular value decomposition (SVD),  orthogonal projection matrix;inverse image index set of a mapping formula, dense subset