The Splaying On Perturbation Of Matrix Eigenvalues And A New Kind Of Relative Efficiency

Matrix perturbation analysis mainly study the effect that the variances of matrix elements influnce 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 marix 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, markov process and so on. As it has important significance and comprehensive application, the eigenvalue problem of marix perturbation is one of research projects which has rich theoretical sense and comprehensive application background. The theory of matrix eigenvalue perturbation gain a 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. This paper has matrix eigenvalue perturbation as main breakthrough point, synchronously contacts location of matrix eigenvalue and perturbation bounds for the unitary polar factor, and give a new kind of definition of relative efficicencies by analysing condition numbers of matrix. The second chapter gives a new Disc theorem basing on the Ostrowski Disc theorem. The third chapter give two groups of perturbation bounds which are adequate for any matrix. Basing on theorem of matrix eigenvalue perturbation, them break out of the restraint which require matrix be especial. Due to discretionary matrix in complex field all have polar decompositions, the fourth chapter states that the perturbation for the unitary polar factor of matrix has important status in practical application. Presently, there are not many representative conclusions, so this chapter gives a new perturbation bounds for the unitary polar factor. The last chapter extends Norm-type Kantorovich Inequality into condition numbers-type, and apply it to studing relative efficicencies of linear model, and get a series of new upper bounds.