几类特殊矩阵的特征值反问题及灵敏度分析

发布时间：2018-05-21 14:09

本文选题：反问题 + 中心对称矩阵　；参考：《内蒙古工业大学》2017年硕士论文

【摘要】：矩阵特征值反问题在物理学、电学和固体力学等方面都有重要应用,因此一直以来都是计算数学领域的一个热门课题.同时,关于矩阵特征值的灵敏度分析,由于其在模型修正、故障诊断和结构控制系统等领域都有重要应用,因此具有重要的理论意义和实用价值.首先,在广义特征值方面,相关成果已有很多.本文利用Hermite广义反Hamilton矩阵的性质和结构特点,通过奇异值分解得到解的一般表达式,并得到了给定矩阵的最佳逼近解.类似地,又得到了广义特征值反问题的中心对称和反中心对称解及其最佳逼近解.其次,在二次特征值反问题方面,分析了自反解存在的条件,并给出了解的一般表达式和最佳逼近解.然后通过将二次特征值反问题转化为广义特征值反问题,得到了一类阻尼系统中的二次特征值反问题的解.最后,在二次矩阵多项式特征值的灵敏度分析中,基于正则矩阵束广义特征值问题亏损特征值的导数得到了二次矩阵多项式亏损特征值的导数.
[Abstract]:Inverse eigenvalue problem of matrix has important applications in physics, electricity and solid mechanics, so it has always been a hot topic in the field of computational mathematics. At the same time, the sensitivity analysis of matrix eigenvalues has important theoretical significance and practical value because of its important applications in the fields of model modification, fault diagnosis and structural control system. First of all, in the generalized eigenvalue, there have been a lot of relevant results. In this paper, by using the properties and structural characteristics of Hermite generalized inverse Hamilton matrix, the general expression of solution is obtained by singular value decomposition, and the best approximate solution of given matrix is obtained. Similarly, the centrosymmetric and anti-centrosymmetric solutions of generalized inverse eigenvalue problems and their best approximation solutions are obtained. Secondly, for the inverse problem of quadratic eigenvalue, the existence conditions of the reflexive solution are analyzed, and the general expression and the best approximate solution of the solution are given. Then, by transforming the inverse problem of quadratic eigenvalue into the inverse problem of generalized eigenvalue, the solution of the inverse problem of quadratic eigenvalue in a class of damping systems is obtained. Finally, in the sensitivity analysis of the eigenvalue of quadratic matrix polynomial, the derivative of the defective eigenvalue of quadratic matrix polynomial is obtained based on the derivative of the defective eigenvalue of the regular matrix bundle generalized eigenvalue problem.
【学位授予单位】：内蒙古工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.6

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