The Singular Value Decomposition And Its Application In The Theory Of Generalized Inverse

The singular value decomposition plays a very important role in matrix theory and numerical linear algebra, especially in the study of generalized inverses of matrices. Professor Musheng Wei has solved many very difficulty problems in matrix theory, by applying the tool of generalized singular value decompositions, see [16, 61, 63, 60, 62, 64, 66]. The main tool in this paper is also generalized singular value decompositions. By applying generalized singular value decompositions, in this paper we obtain the reverse order laws of the least squares g-inverses and minimum norm g-inverses of the products of two matrices, obtain many interesting properties of varies g-inverses of the bordered matrix M=(A B C 0), andstudy the equivalent conditions of block independence of submatrix on g-inversesand least squares g-inverses.This paper is organized as follows:The first part is introduction. In this part, we mainly mention the history and some applications of generalized inverses, development and generality of singular value decomposition, list the main results and the innovation of this paper.In the second part, we study the problems which have been mentioned above.In chapter 1, we reveal the structures of least squares g-inverses and minimum norm g-inverses of the products of two matrices at large by constructing special nonsingular matrix in P-SVD, and finally solve the problem of the reverse order laws of least squares g-inverses and minimum norm g-inverses of the products of two matrices. Meanwhile, we also get a new equivalent condition of the reverse order laws of pseudoinverse of the products of two matrices.In chapter 2, we study the generalized inverses of a bordered matrix. Bordered matrix has very wide applications in the matrix theory, many specialists and scholars had studied bordered matrices before. Based on their research, this paper systemically study the generalized inverses of bordered matrix,including g-inverse, reflexive g-inverses, least squares g-inverses and minimum norm g-inverses. The main tool is QQ-SVD. Especially, in studying least squares g-inverses and minimum norm g-inverses, we also construct two special nonsingular matrices in the QQ-SVD, and obtain the explicit construction of least squares g-inverses and minimum norm g-inverses of bordered matrix. Meantime we also get the explicit construction of pseudo-inverses of bordered matrix and obtain many new results.In chapter 3, we study the block independence of submatrices of a block matrix on generalized inverses. Hall[23] firstly studied the block independence of a block matrix on generalized inverses, then Chen[9,10] and Wang[59] also studied this problem, respectively. In [59], Wang gave a new definition of block independenceand studied the block independence of a block matrix on reflexive g-inverses under his own definition. In this paper, we study the block independence of 2 x 1 and 2x2 block matrices on g-inverses and least squares g-inverses under the definition denoted by [59], and point out the relation between the definition denoted by [23] and [59].