Abstract: By using the concept of partial ordering of matrix, some new definitions of partial ordering were put forward, such as $\boldsymbol{A}\leqslant^{\{1\}}\boldsymbol{B}\Leftrightarrow \boldsymbol{A}\boldsymbol{A}\{1\}=\boldsymbol{B}\boldsymbol{A}\{1\},\boldsymbol{A}\{1\}\boldsymbol{A}=\boldsymbol{A}\{1\}\boldsymbol{B} $ and $\boldsymbol{A}\leqslant^{\{1,2\}}\boldsymbol{B}\Leftrightarrow \boldsymbol{A}\boldsymbol{A}\{1,2\}=\boldsymbol{B}\boldsymbol{A}\{1,2\},\boldsymbol{A}\{1,2\}\boldsymbol{A}=\boldsymbol{A}\{1,2\}\boldsymbol{B}$. Four situations were discussed in detail, according to which, sufficient and necessary conditions of the new partial ordering have been derived.

Key words: generalized inverse, partial ordering, matrix, generalized inverse, partial ordering