含不确定元素的矩阵的特征值计算

In the real world, the uncertainty on the data of a system is a common phenomenon, which causes a significant impact on the system. Since the matrix is an important tool to study the system, the uncertainty on the data of a system leads to the uncertainty on the elements of the system matrix. Motivated by this kind of problems from the real world, we will study the eigenvalue calculations of matrices with uncertainties in this project by the probability method and the interval method, and the corresponding problems in mathematics are called the random eigenvalue problem and the interval eigenvalue problem, respectively. To be specific, we first consider the standard eigenvalue problem and the generalized eigenvalue problem with random coefficient matrices. Then we try to extend the eigenvalue calculus of matrices to that of interval matrices by applying the interval theory. Also, we will study the interval eigenvalue problem with the help of matrix computation theories and optimization theories. Finally we discuss whether there is a link between the statement that the eigenvalue bounds reach at the vertex matrices and the statement that the interval matrix has no overlapping eigenvalue intervals. The expected outcome of our project will not only solve some difficult and valuable problems on random eigenvalues and interval eigenvalues theoretically, but also has an application in the real world due to the widespread use of the random matrix and the interval matrix in statistical and engineering fields.

在生产生活中，系统数据的不确定现象广泛存在，并对系统产生不可忽略的影响。矩阵是描绘系统的重要工具，而系统数据的不确定性导致了系统矩阵元素的不确定性。基于此类实际问题的驱动，本项目将利用概率方法和区间方法研究含不确定元素的矩阵的特征值计算，它们对应的数学问题分别称为随机矩阵特征值问题和区间矩阵特征值问题。具体来说，我们将首先考虑系数矩阵是随机矩阵的标准特征值问题和广义特征值问题；然后尝试运用区间数学理论将矩阵特征值的计算方法推广至区间矩阵的特征值计算；我们还将用矩阵计算和优化等理论解决区间特征值问题；并探索区间矩阵的特征值界在端点矩阵达到与区间矩阵具有不相交特征值区间这两者是否存在联系。本项目预期的研究成果不仅可以从理论上解决随机矩阵特征值问题和区间矩阵特征值问题中的若干具有一定难度和价值的问题，并且由于随机矩阵和区间矩阵在统计和工程领域的广泛应用，预期成果在实际领域也具有应用价值。

本项目主要研究了含不确定元素矩阵的特征值计算。我们研究了对称区间矩阵内部特征值界的估计，以矩阵计算的相关理论为基础，提出了一个循环算法解决该问题，并给出算例验证方法的有效性。我们以区间矩阵正则性的相关充分条件为基础，提出了一种新的计算标准与广义区间特征值问题边界特征值界的方法。与已有方法相比较，该方法花费较少的计算时间。此外，本项目还研究了一类无限维重正倒向随机微分方程，给出了方程解的存在唯一性，得到了解的稳定性。本项目的研究成果对动力系统的稳定性分析、量子力学、图论等领域中的有关问题具有重要意义。