Representation Of Moore-Penrose Inverse Of Matrix Multiplication Perturbation And Applications

The theory of generalized inverse and its applications are one of the very active research directions in the mathematical community.In theoretical research and engineering applications,many practical problems can be reduced to the mathematical problem of solving the generalized inverse of a matrix.For example,it has important applications in the fields of combinatorics and graph theory,Markov chains,numerical analysis,differential equations,cryptography and cybernetics.Schur complement is an important concept that was proposed and studied almost simultaneously with generalized inverse theory.It has a wide range of applications in matrix theory,region decomposition,statistics,and grid theory,and it is also an indispensable and important research tool in many modern research fields.The concept of multiplicative perturbation is relative to additive perturbation,because when we solve the matrix problem,the error between the estimated value and the real value will lead to deviation in the result,so the result can be made more accurate by discussing the analysis and discussion of the matrix before and after the perturbation.And later some scholars found that multiplicative perturbation can be used to calculate the Moore-Penrose inverses of the block matrix.The main research work of this paper is based on this.The representation of the Moore-Penrose inverses of a block matrix has been an important research topic in generalized inverse theory.In this paper,we use the notion of generalized Schur complement and the theory of the representation of the Moore-Penrose inverses of a multiplicative perturbation to derive the Moore-Penrose inverses of a block matrix.And further exploration is carried out for the representation of Moore-Penrose inverses of block matrices.First,the article will introduce some concepts about Moore-Penrose inverses,including the definition,value domain,zero space and other aspects.Then the article will give some major lemmas,such as the Moore-Penrose inverse representation of the sum of two matrices,and some special expressions for the Moore-Penrose inverse of a block matrix.Then the article will introduce the Moore-Penrose inverses representation of the multiplicative perturbation of a matrix and the decomposition of a block matrix into a multiplicative perturbation form using the generalized Schur complement.Finally,the article use the above theoretical knowledge to derive the Moore-Penrose inverse of the block matrix.We mainly generalize the following two results:(?)These two equations are the results obtained using different generalized Schur complements,respectively,and we can give different representations from the above two equations under weaker conditions and subsequently reproduce these two results by strengthening the conditions to generalize and refine the previous work.Finally,the article will give a numerical arithmetic example to prove the correctness and feasibility of the theorem.