Upper Bounds On Eigenvalue Variation

When an element or more elements of a matrix have some small changes, the results of this matrix will be affected, we call this process of change is the matrix perturbation analysis. Matrix perturbation theory is an important component of the matrix eigenvalue perturbation problem. The application of the matrix eigenvalue problem is very extensive. It not only plays an important application in mathematics, such as the numerical calculation, the matrix equation, optimal control theory and nonlinear programming problems; and also plays an important application in physics, and other issues, such as quantum mechanics and computational physics.The purpose of this paper is to study the relative bounds of eigenvalue under the multiplicative perturbation. We get these results which are still valid under slightly more general conditions than the known conditions.The details will go as follows:In Chapter l,we introduce the current study situation which associated with this article and the main content of this paper.In Chapter2, basic knowledge of matrix perturbation analysis and some conceptions and theories which associated with this article in the matrix perturbation analysis are introduced.In Chapter3, the perturbation bounds for the eigenvalues of a diagonalizable matrix under the multiplicative perturbation are investigated. We show that these bounds are still valid under slightly more general conditions and we extend the existing results.