拟三对角矩阵的逆特征值问题
发布时间：2018-08-22 17:52
【摘要】：从上世纪开始科研人员就在陆续讨论根据一些矩阵的特性要素来重组矩阵问题,称之为矩阵的逆特征值问题。逆特征值问题源自于量子力学,分子光特征值学,自动控制及核工程等众多领域的实际应用的需要。逆特征值问题不仅为众多领域的研究提供了一个较为理想的工具和一个较为满意的数学方法,同时,逆特征值问题由于其本身的复杂性与扰动性所带来的一种极为吸引人的数学魅力,使其具有更加广阔的探究前景,吸引了众多学者研究相关课题。目前,研究的矩阵多数情况局限为实数情况,对非零数为复数情况的矩阵的研究尚少。本文的主要内容是对一类特殊形式的复对称矩阵即拟三对角矩阵的逆特征值问题的研究。本文的创新点在于将数域扩大,由研究相对成熟的实数域延拓至复数域,并得到较为理想的结论和较为稳定的算法。下面将本文的主要内容归结为三方面。首先,研究拟三对角矩阵的特征值问题。本文讨论了拟三对角矩阵及其主子阵的特征值情况及其两者的关系,我们得到了拟三对角矩阵的特征值是互异的实数,其主子阵的特征值也是互异实数且两组特征值满足交错性。在这部分内容中将数域和重数细化,使得逆特征值问题的已知条件更具体明晰,给逆特征值问题的研究带来方便。其次,对拟三对角矩阵的逆特征值问题进行探索。在这部分我们从爪形矩阵的构造入手,通过给定的特征值构造爪形矩阵的边界元素,并得到了拟三对角矩阵逆特征值问题存在的充分条件,在保证解存在的前提下,根据爪形矩阵利用酉相似理论构造拟三对角矩阵。最后,给出了拟三对角矩阵逆特征值问题的算法,并给出三个典型的算例,同时借助Matlab对算法进行了大量的数据验证,数据试验表明在满足解存在的条件下,任意给定数据所得到的矩阵符合我们的理想情况,且此算法的稳定性较好。
[Abstract]:Since the last century, researchers have been discussing the problem of reconstructing matrices according to the characteristic elements of matrices. It is called the inverse eigenvalue problem of matrices. Research in this field provides an ideal tool and a satisfactory mathematical method. At the same time, inverse eigenvalue problem has a very attractive mathematical charm because of its complexity and perturbation, which makes it have a broader exploration prospect and attracts many scholars to study related topics. The main content of this paper is to study the inverse eigenvalue problem of a special kind of complex symmetric matrix, i.e. Quasi-Tridiagonal matrix. In this paper, the eigenvalue problem of Quasi-Tridiagonal matrices is studied. In this paper, the eigenvalue of Quasi-Tridiagonal matrices and their principal submatrices and their relations are discussed. We obtain that the eigenvalues of Quasi-Tridiagonal matrices are different from each other. In this part, the number field and multiplicity are refined to make the known conditions of inverse eigenvalue problem more clear, which brings convenience to the study of inverse eigenvalue problem. Secondly, the inverse eigenvalue problem of Quasi-Tridiagonal matrix is explored. In this paper, we start with the construction of claw matrix, construct the boundary elements of claw matrix by given eigenvalue, and obtain the sufficient conditions for the existence of inverse eigenvalue problem of Quasi-Tridiagonal matrix. On the premise of guaranteeing the existence of solution, we construct Quasi-Tridiagonal matrix by using unitary similarity theory according to claw matrix. The algorithm of inverse eigenvalue problem is given, and three typical examples are given. At the same time, a large number of data validation of the algorithm is carried out with the help of MATLAB. The data experiments show that the matrix obtained from any given data conforms to our ideal condition and the stability of the algorithm is good.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O151.21
本文编号：2197859
[Abstract]:Since the last century, researchers have been discussing the problem of reconstructing matrices according to the characteristic elements of matrices. It is called the inverse eigenvalue problem of matrices. Research in this field provides an ideal tool and a satisfactory mathematical method. At the same time, inverse eigenvalue problem has a very attractive mathematical charm because of its complexity and perturbation, which makes it have a broader exploration prospect and attracts many scholars to study related topics. The main content of this paper is to study the inverse eigenvalue problem of a special kind of complex symmetric matrix, i.e. Quasi-Tridiagonal matrix. In this paper, the eigenvalue problem of Quasi-Tridiagonal matrices is studied. In this paper, the eigenvalue of Quasi-Tridiagonal matrices and their principal submatrices and their relations are discussed. We obtain that the eigenvalues of Quasi-Tridiagonal matrices are different from each other. In this part, the number field and multiplicity are refined to make the known conditions of inverse eigenvalue problem more clear, which brings convenience to the study of inverse eigenvalue problem. Secondly, the inverse eigenvalue problem of Quasi-Tridiagonal matrix is explored. In this paper, we start with the construction of claw matrix, construct the boundary elements of claw matrix by given eigenvalue, and obtain the sufficient conditions for the existence of inverse eigenvalue problem of Quasi-Tridiagonal matrix. On the premise of guaranteeing the existence of solution, we construct Quasi-Tridiagonal matrix by using unitary similarity theory according to claw matrix. The algorithm of inverse eigenvalue problem is given, and three typical examples are given. At the same time, a large number of data validation of the algorithm is carried out with the help of MATLAB. The data experiments show that the matrix obtained from any given data conforms to our ideal condition and the stability of the algorithm is good.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O151.21
【参考文献】
相关期刊论文 前2条
1 张振跃;周期Jacobi矩阵的逆特征值问题[J];高等学校计算数学学报;1991年03期
2 王正盛;实对称五对角矩阵逆特征值问题[J];高等学校计算数学学报;2002年04期
，本文编号：2197859
本文链接：https://www.wllwen.com/kejilunwen/yysx/2197859.html
