一类新型结构矩阵和结构矩阵束特征问题的算法及其应用

Eigenproblems of structured matrices and structured matrix pencils have been an active field of research in numerical algebra during the last two decades, which derive from many applied disciplines such as physics, mechanics, and control theory, etc. The merits of structured preserving algorithms for computing the eigenproblems of structured matrices or structured matrix pencils lie in the good efficiency of the structured preserving algorithms. Moreover, the structured preserving algorithms remain the practical physics significance owning to the spectral of matrices or matrix pencils. The topics for this project arise from numerical algebra problems in quaternionic quantum mechanics, and the main purposes are studying the eigenproblems of structured quaternion matrices and structured quaternoin matrix pencils by applying the theory of structured matrices. Furthermore, the applications of the above eigenproblems in solving mathematical models in quaternionic quantum mechanics and control system will be considered. The real representation matrix of a general quaternion matrix is a novel type of structured matrix with JRS-symmetric structure. Based on this structure, the goals of this project continue to research the structure of real representation of structured quaternion matrices and structured quaternion matrix pencils, discuss their algebraic properties, characterize their canonical forms by constructing novel structure preserving transformations, design structure preserving algorithms for the eigenproblems of structured quaternion matrics and sturctured quaternion matrix pencils, and analyze the convergency. At last, numerical experiments will be provided to demonstrate the efficiency of the structure preserving algorithms.

结构矩阵和结构矩阵束的特征问题产生于物理、力学、控制理论等应用学科，是近二十年来数值代数领域研究的热点问题。利用保结构算法计算结构矩阵或结构矩阵束的特征问题不但非常有效而且能够保持矩阵或矩阵束的谱信息本身所代表的实际物理意义。本项目研究的课题来源于四元数量子力学中的数值代数问题，旨在用结构矩阵的理论研究结构四元数矩阵和结构四元数矩阵束的特征问题，以及它们在四元数量子力学和控制系统数学模型求解中的应用。一般四元数矩阵的实表示矩阵是一种具有JRS-对称结构的新型结构矩阵，本项目将在此基础上，继续深入探讨结构四元数矩阵和结构四元数矩阵束实表示的结构特征，研究它们的代数性质，构造新的保结构变换刻画它们的标准形，设计求解结构四元数矩阵和结构四元数矩阵束特征问题的保结构算法并进行收敛性分析，用数值试验验证算法的有效性。

本项目在执行过程中，按照预定研究计划，紧紧围绕四元数矩阵实表示矩阵的JRS-对称结构，以研究四元数矩阵特征值问题为中心，刻画了四元数矩阵的实表示和复表示矩阵特征值之间的关系，形成了一套研究四元数矩阵特征值问题的代数方法和数值算法。以研究此类问题为出发点，研究并突破了一系列与四元数矩阵特征值问题密切相关的结构矩阵的计算和应用问题，取得了丰富的研究成果，相信能为四元数应用学科的发展和结构矩阵的研究提供理论支持和推动力。发表与项目相关的SCI检索科研论文8篇，达到了预期研究目标。项目执行期间毕业了2名研究生，项目组成员参加在国内举办的国际会议2人次，国内会议2人次。邀请国内外高水平学者来访5人次。