【作者】 张妍；

【导师】 王兴涛；

【作者基本信息】 哈尔滨工业大学 ， 运筹学与控制论， 2020， 硕士

【摘要】 矩阵的逆特征问题是指由给定的特征值或特征向量构造出相应矩阵的问题。逆特征值问题在实际工程技术里应用较多,加之问题本身的数学魅力,使其研究具有非常广阔的前景,日益成为数学中一个十分活跃的课题。随机矩阵是一类特殊且应用广泛的非负矩阵,它在有限齐次马尔科夫链理论、组合数学,生物及社会科学中的进化系统模型和离散经济模型等机模型中都有重要应用。本文对低阶对称双随机矩阵逆特征问题进行了讨论,并且对高阶对称双随机矩阵的逆特征值问题提出了一种构造方法。对于低阶双随机矩阵的逆特征问题,给出了低阶双随机矩阵逆特征值问题解存在的充分条件和矩阵的构造形式。使低阶对称双随机矩阵的逆特征值问题的判断更加简单,应用更加简便。给出的具体例子,说明了应用的可行性和简便性。对于高阶双随机矩阵的逆特征值问题,证明了由两个对称双随机矩阵合并成更高阶对称双随机矩阵的结论。也就是说,把利用已知谱的较小矩阵构造新的对称双随机矩阵的方法用于高阶对称双随机矩阵逆特征值问题,并得出了高阶双随机矩阵逆特征值问题解存在的充分条件,避开了讨论特征值个数的奇偶性。通过实例将这一结果做了应用。更多还原

【Abstract】 The inverse eigenvalue problem of matrices refers to the problem of constructing a corresponding matrix from given eigenvalues or eigenvectors.The inverse eigenvalue problem is widely used in practical engineering technology,and the mathematical charm of the problem itself makes its research have a very broad prospect and become a very active topic increasingly in mathematics.A stochastic matrix is a kind of special and widely used nonnegative matrix.It has important applications in the finite homogeneous Markov chain theory,combinatorial mathematics,evolutionary system model and discrete economic model in biological and social sciences.In this thesis,the inverse eigenvalue problem of low-order symmetric double stochastic matrices is discussed,and a construction method for the inverse eigenvalue problem of high-order symmetric double stochastic matrices is proposed.On the basis of predecessors,some results of inverse eigenvalue problem of symmetric double stochastic matrices are improved,and the conclusion is convenient for application.The sufficient conditions for the existence of symmetric double stochastic matrices are obtained.The construction of a solution matrix of the inverse eigenvalue problem of low order double stochastic matrices is given.It makes the judgment of the inverse eigenvalue problem of low order symmetric double stochastic matrices simpler and the application more convenient.Examples are given to illustrate the feasibility and simplicity of the application.For the inverse eigenvalue problem of higher-order symmetric double stochastic matrices,the conclusion that two symmetric double stochastic matrices are combined into higher-order symmetric double stochastic matrices is proved.In other words,the method of constructing a new symmetric double random matrix by using the smaller matrix of known spectrum is applied to the inverse eigenvalue problem of higher-order symmetric double stochastic matrix.This method avoids discussing the parity of the number of eigenvalues.The results are applied to practical examples.更多还原