Perturbation Bounds For Matrix Eigenvalues

The purpose of this paper is to study the additive perturbation of matrix eigenvalue. Some new absolute and relative perturbation bounds of matrix are obtained.Matrix perturbation analysis mainly studies the effect that the variances of matrix elements influence the sequence of matrix. It is not only relevant with theory of matrix and theory of operator, but also is important to matrix count.The eigenvalue problem of matrix perturbation not only cope to problem of mathematic count, such as linear programming, optimization, differential equations, but also have important applications in structural mechanics, control design, computational physics and quantum mechanics. Presently, in most cases, the matrix eigenvalue is applied in solving the equation of mathematical physics, difference equation, and Markov process and so on. As it has important significance and comprehensive application, the eigenvalue problem of matrix perturbation is one of research projects which has rich theoretical sense and comprehensive application background.The theory of matrix eigenvalue perturbation gains an adequate development in the latter half of the last century. Overseas systems are relative perfect, and establish the basic framework of the theory of matrix eigenvalue perturbation. Since the mid-eighties of the last century, a batch of domestic academician who devote to basic study, have made great strides. At analytical method and field of research of the theory of matrix eigenvalue perturbation, there are quantum jumps which would have oriented and quotable effects when be applied to other subjects.The details will go as follows:Firstly the current study situation and basic knowledge of matrix eigenvalue are introduced.Secondly we improve some previous corresponding results in some sense.Using the singular value decomposition, Some new Wielandt-Hoffiman type and Weyl type absolute perturbation bounds of special matrix, symmetrical matrix diagonalizable matrix are obtained. We also analyze the arbitrary perturbation of normal matrix, based on we give Weyl type absolute perturbation bounds of normal matrix. Using the schur triangular factorization of matrix, we extend the absolute perturbation bounds of arbitrary matrix. We use some new methods to get results, which improve and extend the corresponding results in other Papers.Lastly, the new relative perturbation bounds for special matrix and diagonalizable matrix are obtained. Using the schur triangular factorization of matrix, we deal with the relative Perturbation bounds of arbitrary matrix and give some new theorems.