Research On Generalized Singular Value Decompositions And Its Applications With Respect To Semi-definite Inner Product

This paper mainly discusses the two kinds of generalized singular value decomposition. One is the weighted singular value decomposition, the other is the generalized singular value decomposition with respect to a semi-definite inner product. As we all know, the singular value decomposition is very important tool both in theoretical research of numerical linear algebra, and in the numerical calculation. Therefore, the research of theory and calculation of singular value decomposition has a very important significance research in other fields.The traditional weighted singular value decomposition of two matrices are symmetric and positive definite,in this paper, we relax one of weight matrix is symmetric positive semi-definite,i.e. combine with the theory of generalized inverse of a matrix study on the weighted singular value decomposition with one of the weight matrices being positive semi-definite and the other being positive definite.For the generalized singular value decomposition with respect to a semi-definite inner product,we mainly use the methods of singular value decomposition of matrices in a space with an indefinite scalar product and two-sided hyperbolic singular value decomposition. Study the generalized singular value decomposition with respect to a semi-definite inner product, and the two-side generalized singular value decomposition with respect to a semi-definite inner product.