具有指定代数重数和几何重数下界的特征值的最邻近结构矩阵

中文摘要 AI 辅助

我们研究了一个问题：将属于线性结构实矩阵的n阶子空间S中的给定矩阵X扰动到其最邻近的矩阵Y=X+D，其中D属于S，且Y具有一个特征值，该特征值的代数重数和几何重数满足给定的下界。所提出的框架同时处理了目标特征值已知的情况，以及目标特征值被视为未知决策变量、需与结构扰动共同确定的情况。我们通过用结构坐标中的Jordan链关系表达可行性约束，建立了此类矩阵存在的充分必要条件。这使得我们能够将该问题表述为一个非线性约束嵌套优化问题，该问题最小化扰动的Frobenius范数。为解决此问题，我们实施了一种两层策略，利用MATLAB的fmincon通过多起点序列二次规划方法进行连续内层优化，而外层则求解一个离散优化问题。对各类结构矩阵进行的全面数值实验，包括与现有方法的比较，证明了所提出方法的有效性和准确性。

英文摘要

We study the problem of perturbing a given matrix X belonging to a subspace S of linearly structured real matrices of order n to its nearest counterpart Y=X+D, where D belongs to S and Y possesses an eigenvalue with prescribed lower bounds on its algebraic multiplicity and geometric multiplicity. The proposed framework takes care of both cases where the target eigenvalue is known and where it is treated as an unknown decision variable to be determined jointly with the structured perturbation. We establish necessary and sufficient conditions for the existence of such matrices by expressing the feasibility constraints through Jordan-chain relations in structural coordinates. This allows us to formulate the problem as a nonlinear constrained nested optimization that minimizes the Frobenius norm of the perturbation. To solve this, we implemented a two-level strategy, utilizing MATLAB's fmincon for continuous inner optimization via a multi-start Sequential Quadratic Programming approach, while the outer level solves a discrete optimization problem. Comprehensive numerical experiments on various classes of structured matrices, including comparisons with existing methods, demonstrate the effectiveness and accuracy of the proposed approach.

发表机构

• Mizoram University（米佐拉姆大学）
• Indian Institute of Technology Delhi（德里印度理工学院）

机构由 AI 辅助整理，请以论文原文为准。