| The representation of the Drazin inverse of block matrix is a classic problem in the study of the Drazin inverse: let be a 2 × 2 block complex matrix,and give the explicit expression of the Drazin inverse of matrix by sub-block of the matrix and the Drazin inverse of each diagonal sub-block.T he representation of the Drazin inverse of 2 × 2 block matrices has a profound application backgroun d in differential equations,automation,numerical analysis,economics,cybernetics and many other as pects.This paper mainly studies the representation of the anti-triangular block matrix Drazin inverse a nd applies it to the resulting representation of the Drazin inverse of the 2 × 2 block matrix.The first chapter introduces the relevant basic knowledge and the current research status at home and abroad,a nd summarizes the main research contents of this paper.The second chapter lists the lemmas and basi c knowledge required in the calculation of the article,which plays a key role in the solution and proof of the theorems.In the third chapter,firstly,the conditions for finding the Drazin inverse of anti-tria ngular block matrix are obtained by matrix operation on the field of complex numbers,and the form ulae for the Drazin inverse of anti-triangular block matrix are established under this new set of condit ions.Secondly,as an application,the expressions of the Drazin inverse of 2 × 2 block matrix are esta blished by using the results of sub-blocks of 2 × 2 block matrix and the Drazin inverse of the abovementioned inverse triangular matrix.These conditions and results generalize a series of results about the Drazin inverse in literature.In chapter 4,the generalized Drazin inverse of anti-triangular block o perator matrix is calculated by using new conditions in Banach space,and these results are extended t o the generalized Drazin inverse of 2 × 2 block operator matrix.Chapter 5 illustrates the generalizati on and practicability of the results through some numerical examples. |
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