Some Generalized Results About Wilkinson Theorem

Wilkinson theorem is a classical result on perturbation analysis of algebraic eigenvalue problem, it is a very important tool especially when studying the sensitivity analysis of matrix eigenvalues. This thesis mainly discuss the relationship between the condition number of matrix eigenvalue problem and the shortest distance from that problem to the ill-posed set, and finally generalize Wilkinson theorem to some common eigenvalue problems, including the matrix eigenvalue problems of periodic pairs, skew-Hamiltonian matrix, matrix polynomial and multiparameter system. This thesis consists of six parts.In section 1, the main content of Wilkinson theorem and some results concerning it are referred, then we point out several important implications of Wilkinson theorem in other research fields.In section 2, we study and show several important results deriving from Wilkinson theorem and some other related tools. These results are the basic tools for the following research.Sections 3 to 6 are the main parts of this thesis. In section 3, we discuss the eigenvalue problem of periodic pairs. Firstly, we cite some existing results, then generalize Wilkinson theorem to the eigenvalue problem of periodic pairs by some relative proving techniques. By some new techniques, in section 4, 5 and 6, through discussion, we get several generalized results about Wilkinson theorem concerning the eigenproblems of skew-Hamiltonian matrix, matrix polynomial and multiparameter system.