The Study On Generalized Inverse Of Partioned Matrix

The generalized inverse theory of the matrix has been a very important discussion branch in the field of world matrices theory,and it plays a role in solving the general solu-tion,the least square solution of the linear equations and optimized control in engineering operations.We usually use the idea of dividing the matrix into2 x 2 block matrix to study the generalized inverse.After extensive research by scholars,the expression of the gener-alized inverse of the block matrix is diverse,but they have no advantage for calculating the MP-inverse of the general digital matrix.In this paper,we give a new expression of the generalized inverse of the block matrix by generalized inverse additivity and the nature that if the rank of the two matrices is zero,the two matrices are equal.At first,by extending the correlation properties of matrices,we obtain the 2 x 2 block matrix MP-inverse expression with three and two zero sub-blocks.On this basis,we use generalized inverse additivity to obtain the MP-inverse expression of a block matrix with a zero sub-block and without zero sub-blocks.Secondly,we study the equivalence between the Banachiewicz-Schur generalized in-verse form and {1}-inverse,{1,2}-inverse,{1,3}-inverse,{1,2,3}-inverse,{1,4}-inverse,{1,2,4}-inverse of a matrix,provide a new idea for the expression of the d-ifferent generalized inverses of the matrix.We extende the conclusions to the Hermit space,and get the equivalence between them.At last,we study the equivalence between the Banachiewicz-Schur weighted generalized inverse form and its {1,3X}-weighted inverse,{1,2,3X}-weighted inverse,{1,4Y}-weighted inverse,{1,2,4Y}-weighted inverse.At last,we present the MP-inverse expression of bordered matrix based on this de-composition and get a new idea for solving the generalized inverse of general digital ma-trix.