This manuscript proposes a generalized inverse for a dual matrix called dual Drazin generalized inverse (DDGI) which generalizes the notion of the dual group generalized inverse (DGGI). Under certain necessary and sufficient conditions, we establish the existence of the DDGI of a dual matrix of any index. Thereafter, we show that the DDGI is unique (whenever exists). The DDGI is then used to solve a linear dual system. We also establish reverse-order law and forward-order law for a particular form of the DGGI, dual Moore-Penrose generalized inverse (DMPGI), dual core generalized inverse (DCGI), and DDGI under certain suitable conditions. Finally, the partial-orders based on DCGI and DGGI are proposed.

翻译：本文提出了一种对偶矩阵的广义逆——双重Drazin广义逆（DDGI），该逆推广了对偶群广义逆（DGGI）的概念。在必要的充分必要条件下，我们建立了任意指标对偶矩阵的DDGI存在性定理。随后证明了DDGI在存在时具有唯一性，并将其用于求解线性对偶系统。我们还针对DGGI、对偶Moore-Penrose广义逆（DMPGI）、对偶核心广义逆（DCGI）及DDGI的特定形式，在适当条件下建立了反序律与正序律。最后，基于DCGI与DGGI提出了偏序关系。