【作者】 贾丽杰；

【导师】 杨虎；

【作者基本信息】 重庆大学 ， 应用数学， 2005， 硕士

【摘要】 矩阵扰动分析主要是研究矩阵元素的变化对于矩阵问题的解的影响,它不仅仅与矩阵论和算子理论密切相关,而且对于矩阵计算同样是有重要的意义。矩阵扰动中的矩阵特征值问题不仅可直接解决数学中诸如非线性规划、优化、常微分方程,以及各类数学计算问题,而且在结构力学、工程设计、计算物理和量子力学中具有重要作用,目前矩阵特征值问题的应用大多来自于解数学物理方程、差分方程、Markov 过程等。正因为它具有重要意义和广泛的应用,所以矩阵特征值扰动问题是具有深度理论意义和广泛应用背景的研究任务之一。矩阵特征值扰动理论在上个世纪后半叶得到充分的发展,国外的发展体系比较完善,建立了矩阵特征值扰动理论的基本框架,国内在上世纪80 年代中期以后,一批致力于基础理论研究的数学工作者在这一领域取得了长足的发展,使矩阵特征值扰动理论的分析方法、研究范围都有突破性的进展,为其在其他学科上的应用起到了导向和借鉴作用。本篇文章以研究矩阵特征值扰动为主要的问题切入点,同时联系矩阵特征值定位问题和矩阵酉极因子扰动,并且由对矩阵条件数的分析给出一类新的相对效率的定义。本文第二章在Ostrowski 圆盘定理的基础上给出新的圆盘定位定理。第三章,在已有矩阵特征值扰动定理的基础上,突破对矩阵为特殊矩阵的束缚,给出两组新的适用于任意矩阵的扰动上界。在第四章,由于复数域上的任意矩阵都可以进行酉极因子分解,所以矩阵的酉极因子的扰动在实际应用中同样占有重要位置。目前具有代表性的结果不是很多,本章中给出新的酉极因子扰动欧氏范数上界。最后一章将目前的范数类下的广义康氏不等式拓展到条件数类,并且将其应用到线性模型的相对效率的研究之中,得出一系列新的上界。更多还原

【Abstract】 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.更多还原