The Drazin Inverse Of The Sum Of Two Matrices And Its Applications

Drazin inverse of a square matrix is widely applied in many fields, such as singular differential and difference equations, Markov chains, operator theory, iterative method, cryptography and numerical analysis, etc. At the same time, the Drazin inverse in perturbation bounds for the relative eigenvalue problem has important application. Therefore, since the middle of last century, the Drazin inverse problem of matrix has become an important area of research, up to now those problems have caused a lot of people’s interests in the world.Firstly, the introduction part of the article mainly describes the research background and significance of the Drazin inverse of matrix, and the final direction of the Drazin inverse of matrix is pointed out. With the Drazin inverse of matrix in an increasingly important role in various disciplines, the Drazin inverse expression research of the sum of two matrices has played an important role in promotion. This paper studies the new conditions, the sum of two matrices, and the Drazin inverse of partitioned matrix expression.Secondly, based on the core-nilpotent decomposition of matrix, we give the results for the Drazin inverse of P+Q under the new contidions. At the same time another relative symmetry condition is given. We gain different results, and then derive a representation of the Drazin inverse of a block matrix M=A B C D under some conditions. Moreover, some alternative representations for the Drazin inverse of MD where the generalized Schur complement S= D-CADB is nonsingular.The group inverses as a special Drazin inverse, but not all square matrix has its group inverse. In the fourth part of this article, we do the research on the group inverse problem of P+Q in complex fields, derive the existence and expression of the group inverse of block matrices, using the definition of group inverse and the given conditions, at the result of working out the conditions of the group inverse for anti-triangular block matrices under some conditions.Finally, corresponding numerical examples are given to illustrate the expression of the Drazin inverse of matrix on the calculation of superiority.