A Study For Some Matrix Perturbation Problems

In this thesis we investigate perturbation analysis of some matrix computational prob-lems. Applying algebra and elementary calculus techniques, some new perturbation boundsare presented for a variety of matrix computational problems, including the generlized polardecomposition, eigenspace and singular space, Rayleigh quotient for singular values, generalizedeigenvalue problems, nonlinear matrix equation and structured linear systems and so on.In Chapter 1 we introduce some important conceptions in the matrix perturbation analysis,and give overview of our results.In Chapter 2 some perturbation bounds of the generalized polar decomposition for thematrices with different ranks are given. For the subunitary polar factor some former results areimproved. For the Hermitian positive semidefinite polar factor, some perturbation bounds areobtained under any unitarily invariant norm. In particular, the classical result of Hermitianpositive definite polar factors is improved for the Frobenius norm.In Chapter 3 we condiser the additive and multiplicative perturbations of eigenspaces.Some absolute and relative bounds are presented by using double gaps of eigenvalues, whichare better than the existing bounds in some sense.In Chapter 4 the absolute and relative perturbations of singular spaces are investigated forthe Frobenius norm. Using double gaps of singular values, some separate and joint perturbationbounds for left and right singular subspaces are obtained. Moreover, the upper bounds of theircondition numbers are derived. At the same time a relative bound for the Wedin's sin(?) typebound and both right and left singular subspaces relative bounds are given, respectively.In Chapter 5 we consider the Rayleigh quotient of singular value problems. Some results ofthe Rayleigh quotient of Hermitian matrices are extended to those for arbitrary matrix. On theothe hand, some unitarily invariant norm bounds for singular values are presented for Rayleighquotient matrices. Our results improve the existing bounds.In Chapter 6 the generalized eigenvalue perturbation bounds for diagonalizable pairs areinvestigated. Three results of Sun on normal pencils are extened to diagonalizable pencils.In Chapter 7 we consider the exsitence and perturbation analysis of Hermitian positivedefinite solution for X+A^*X^-1A=P. Firstly, the existence of the Hermitian positive definitesolutions is proved and the first order perturbation bound of the maximal solution is presentedby using differetial method for the matrix equation X+A^*X^-1A=P, which improves thecorresponding partial results in; Secondly(some new perturbation bounds for Hermitianpositive definite solutions of the matrix equations X±A^*X^-1A=I are derived by usingelementary calculus techniques and the new results are illustrated by numerical examples.In Chapter 8 the structured backward errors for solving structured linear systems arediscussed, we present a computable formulae of the structured backward error for two kinds ofstrutured linear systems.