Study On Perturbation Of Eigenvalues Of Matirx

In some large-scale numerical calculation problems , such as engineering calculation , physics,astronomy and numerical solution to differential equations, matrix eigenvalue problem plays an extremely important role. For a given matrix, if you want to use the computer to derive its eigenvalues, however, because of the existence of floating-point error and measurement error, it must exist deviation between the exact value and the numerical solution of eigenvalue, therefore , estimating the reliability of the calculation is one of the important issues to do some research to the matrix perturbation analysis .In this paper, the absolute additive perturbation and relative additive perturbation of matrix eigenvalue in invariant subspaces are studied, and some new perturbation bounds are given. The traditional three types of error bounds: Weyl, Hoffman-Wielandt, and Bauer-Fike, are systematically studied in this article and doing some further study to several special cases, then strengthened the corresponding results.In chapter three, some research about the Weyl type are studied. Considering the existing results of the original matrix A is the Hermite matrix and its perturbation matrix B is normal matrix , or the original matrix A and its perturbation matrix B both are Hermite matrix , some study to the relative perturbation bounds of the Weyl type , under the conditions of a positive semi-definite Hermite matrix.,which has the basis of some perturbation bounds under the conditions of a positive definite Hermite matrix.In chapter four, some research of the Hermite-Wielandt type are studied. Considering the exsisting results of the normal matrix and the diagonal matrix , some research about the normal Hoffman-Wielandt type of normal matrix are done, and some exsit perturbation bounds are improved , some perturbation bounds are easy .In chapter five, some research of the Bauer-Fike type are studied. A relative perturbation bound are given , which has the basis of its absolute perturbation bound.In chapter six, some research of the AQ-QB=R type are studied. Through the use of sub-block matrix, as well as the expansion of matrix. a easy result are given ,and at the same time improving the exsit results,. In the end , some comparison are given between the new and the old results.