某些分块矩阵的Drazin逆

Drazin inverse for some partitioned matrices

分块矩阵的广义逆不仅在数学理论上有广泛研究,而且在自动化、系统控制、概率统计、数学规划等领域有着广泛的实际应用背景,尤其是在最小二乘问题,病态线性、非线性问题,不适定问题,回归、分布估计、马尔可夫链等统计问题,随机规划问题,控制论和系统识别问题等研究中广义逆更是发挥着重要的作用.但求任意2×2分块矩阵的Drazin逆表达式是一个未解决的问题,因此给出了分块矩阵[EED EED E 0],[EED ED E 0],[ED EED E 0],[ED ED E 0]的Drazin逆表达式,其中E为复数域上的方阵,ED为E的Drazin逆.

Abstract ： Generalized inverse of partitioned matrices are not only well developed in mathematic theory, but also ap- plied extensively in practice, including areas such as automation, system control, probability statistics, mathemati- cal programming, and so on. They play a particularly important role in a wide range of statistical problems ranging from least squares, morbidity linearity, non-linearity, ill-posed problem, regression, estimation of distribution, Markov chains, stochastic programming, cybernetics and system identification. But it is still an unsolved problem to find the representation for the Drazin inverse of an arbitrary 2 × 2 partitioned matrix. Herein we gave the represen-tations of the Drazin inverse of a class of partitioned matrices[EE^D EE^D E0],[EE^D E^D E 0],[E^D EE^D E 0],[E^D E^D E 0] Where E is a square matrix over the field of complex numbers, and E^D is the Drazin inverse of E.