The Inequalities Of The Singular Values And Eigenvalues Of Summation And Difference Of Matrices

The studying of the inequalities of eignvalues and singular values is the main issue of the theory of matrices and perturbation analysis, which could be divivded into three main aspects: (1), the relations of the eignvalues and singular values of single matix, (2),the relations of eignvalues and singular values of several matrices, (3), the estimation of their bounds. Early, Wely,Courant,Fisher,Wielandt and Hoffaman have already done many works in these issues. In 1953, A.J.Hoffaman and H.W.Wielandt proposed the famous Wielandt-Hoffaman theorem, which described the ralations of eignvalues of the difference of two symmatric matrices. Concretely, it associated the purturbation of eignvalues with the Enclid norm of the purturbation matrices, and it was the best conclusion for the error estimation which based on the float operation of the orthonormal transform.Some interesting results have been obtained by[9],[12]. In the paper we strengthen H-W's and wu's results, considerably, we give their inver inequation and their convenient proof, then we extend it to the inequality of singular values of arbitrary matrices.Four sections are folded in this thesis. section 1 gives us an introduction of the basic conception of H-W theorem and looks back the former works of this area. In section 2, we briefly introduce the classical H-W theorem and some other definitions. We discuss the symmetric form of H-W theorem and generalise it to the summation of matrix in section 3. The last part present the singular inequalities for the random matrices which include the difference and summation of the matrices and the numerical experiments are presented.