Drazin Inverse Of Perturbation Theory And The Pseudo-drazin Inverse Of Condition Number

The Drazin inverse is very useful in various applications such as in singular differential and difference equations, Markov chains, iterative methods and numerical analysis when the definition of Drazin inverse has been given. Especially, the generalized matrix is extremely important in singular linear ordinary differential equations systems and the solution of singular difference equations systems.The theory that the inverse of a nonsingular matrix is continuous function of the elements of the matrix was established by J. H. Wilkinson. The continuity of the generalized inverse A⁺ of a matrix A was introduced by G. W. Stewart. In this paper, at first, the continuity of the special matrices inverse, such that M -matrices and H -matrices, respectively, are provided. Campbell and Meyer also established the continuity properties of Drazin inverse, but the explicit bound was not given.The Drazin inverse is unstable with respect to perturbation. However, under some specific perturbation E, the closeness of the matrices (A+E)^D and A^D can be proved and the explicit bound the relation error can also be obtained. Based on the different representations of Drazin inverse, many scientists and scholars have worked it research. U. G. Rothblum gave the following representation of Drazin inverse: A^D=(A-H)^-1(I-H)=(I-H)(A-H)^-1, where H=I-AA^D=I-A^D A. Based on the representation, we also obtain the norm estimate of‖(A+E)^D-A^D‖₂/‖A^D‖and‖(A+E)^#-A^D‖₂/‖A^D‖₂ and compare with the precedent results.Various normwise relative condition numbers are key elements that measure the sensitivity of a matrix inverse. In Desmond J. Higham, he stated the main results: various normwise relative condition numbers that measure the sensitivity of matrix inversion and the solution of linear systems are characterized. Wei Yimin, Wang Guanglin and Wang Dingkun have studied various normwise relative condition numbers of Drazin inverse and their condition numbers of singular linear systems. Moreover, in the same year, Wei Yimin and Xu Wei gave condition number of Bott-Duff inverse and their condition numbers. In this paper, on one hand, we give the definition of Pseudo-Drazin inverse just as the Drazin inverse in some aspects, but there exists apparent difference.On the other hand, some properties are studied such as the characterization of the Pseudo-Drazin inverse and the definition of‖·‖_PD, etc. At the same time, various normwise relative condition numbers that measure the sensitivity of the Pseudo-Drazin inverse and the solution of singular symmetrical linear systems are characterized.