Study On Some Problems Of Generalized Inverse Of Matrix Matrices

This paper firstly analyzes the present development situation of the matrix theory on skew field on the whole, and expounds the difficulties of matrix research on skew field, and then lays emphasis on specific research on the generalized inverse matrix of skew field from several parts. The paper by listing some common matrix decomposition of matrix theory over skew field, a better understanding of the nature of the generalized inverse matrix, summarizes and analyzes the Moore-Penrose inverse, the Drazin inverse, the group inverse, etc. And it introduces the Drazin inverse and the group inverse of the block matrix over the skew field and the computing method of the generalized inverse quaternion matrix. Finally the article makes a brief introduction to results and outlook. Generalizing the computing method of the generalized inverse matrix over the complex field to the skew field, we get some theoretical significance and application value of the conclusion of the generalized inverse matrix over skew field. The article is divided into five chapters:The first chapter mainly introduces the research background, research status and development trend of the skew field, the quaternion, the quaternion matrix over the skew field, and then expounds the basic knowledge and basic properties. We also summarize research situations and contribution on the theory of matrix over skew field of mathematics workers in our country. In the end we illustrate the work done in this paper.The second chapter by listing some common matrix decomposition of matrix theory over skew field, a better understanding of the nature of the generalized inverse matrix, summarizes and analyzes the Moore-Penrose inverse, the Drazin inverse, the group inverse, etc.The third chapter mainly introduces the Drazin inverse and the group inverse of the block matrix over skew field. Firstly applying the index of matrix and the property of the group inverse, we give the new demonstrations of the Drazin inverse representation of the upper triangular matrix over skew field, and then obtain some new result of the Drazin inverse formula of sum of the two matrices over skew field. Using the method of the block matrix, we study the existence and representation of the group inverse of the product of the two matrices, then give the sufficient and necessary condition and representation of the existence of the group inverse of the product of the two matrices over skew field. Finally we get the existence and representation of the group inverse of the block matrix over skew field.The fourth chapter introduces the computing method of the generalized in-verse quaternion matrix. Generalizing the computing method of the generalized inverse matrix over the complex field to the skew field, we get two computing methods of computing the generalized inverse matrix over the skew field. The one is using the line left elementary transformation to compute the 1-inverse and 1,2-inverse of the quaternion matrix, the another one is applying the full rank factorization of the quaternion matrix to get the generalized inverse matrix. And we give the example of the computing method. Finally we get the algorithm for inverse matrices of symmetric r-circulant matrices over skew field using the elementary transformation of the polynomial matrix.The fifth chapter makes a brief introduction to results and outlook.