The main contribution of this paper is to present new generalized inverses as weaker versions of a G-outer inverse. In particular, we define and characterize left and right G-outer inverses of rectangular matrices. Solvability of matrix equation systems as AXA = AEA and BAEAX = B; or AXA = AEA and XAEAD = D, where $A\in {\mathbb{C}}^{m\times n}$, $B\in {\mathbb{C}}^{p\times m}$, $D\in {\mathbb{C}}^{n\times q}$ and $E\in {\mathbb{C}}^{n\times m}$, is studied by means of left and right G-outer inverses. The general solution forms of these systems give descriptions of the sets of all left and right G-outer inverses. Using left and right G-outer inverses, we introduce new partial orders and establish their relations with minus partial order and space pre-order. We apply these results to present and investigate left and right G-Drazin inverses of square matrices and corresponding partial orders.

本文的主要贡献是将新的广义逆作为 G 外逆的较弱版本。特别是，我们定义和表征了矩形矩阵的左右 G-outer 逆。矩阵方程组的可解性为AXA  =  AEA和BAEAX  =  B；或AXA  =  AEA和XAEAD  =  D，其中$\mathrm{一个}\in {\mathbb{C}}^{米\times n}$,$乙\in {\mathbb{C}}^{p\times 米}$,$D\in {\mathbb{C}}^{n\times q}$和$乙\in {\mathbb{C}}^{n\times 米}$, 通过左右 G-outer 逆进行研究。这些系统的通解形式给出了所有左右 G 外逆逆集的描述。使用左右 G-outer 逆，我们引入了新的偏序，并建立了它们与负偏序和空间预序的关系。我们应用这些结果来呈现和研究方阵的左右 G-Drazin 逆和相应的偏阶。