摘要
In order to study the Drazin invertibility of a matrix with the generalized factorization over an arbitrary ring, the necessary and sufficient conditions for the existence of the Drazin inverse of a matrix are given by the properties of the generalized factorization. Let T = PAQ be a square matrix with the generalized factorization, then T has Drazin index k if and only if k is the smallest natural number such that Ak is regular and Uk（Vk） is invertible if and only if k is the smallest natural number such that Ak is regular and Uk（Vk） is invertible if and only if k is the smallest natural number such that Ak is regular and Uk（Vk） is invertible. The formulae to compute the Drazin inverse are also obtained. These results generalize recent results obtained for the Drazin inverse of a matrix with a universal factorization.
为了研究任意环上具有广义分解的矩阵的Drazin可逆性,利用广义分解的一些性质,给出了任意环上具有广义分解的矩阵的Drazin可逆的充分必要条件:设T=PAQ为具有广义分解的矩阵,则T的Drazin指标为k当且仅当k为使得Ak正则且Uk(Vk)可逆的最小自然数当且仅当k为使得Ak正则且U珟k(珟Vk)可逆的最小自然数当且仅当k为使得Ak正则且k(Vk)可逆的最小自然数.同时给出了几种计算Drazin逆的公式,推广了任意环上具有泛分解的矩阵Drazin逆的结果.
基金
The National Natural Science Foundation of China(No.10571026,10871051)
Specialized Research Fund for the Doctoral Pro-gram of Higher Education(No.20060286006,200802860024)
