AI 中文总结
本文提出基于黎曼优化的方法求解最接近正规矩阵问题，刻画其理论性质，开发改进的黎曼信赖域算法并通过大量实验验证，可处理更大规模矩阵。
AI 中文摘要
我们提出一种基于黎曼优化的方法，用于计算与给定矩阵最接近的正规矩阵。该问题可表述为在大小为n的酉矩阵流形U(n)或旗流形U(n)/U(1)^n上对光滑函数进行最小化。旗流形特别适用于理论分析：我们刻画了目标函数的全局最大值，并证明对于一般输入，其局部极小值点有限且孤立；由此可得，原最近正规矩阵问题在一般情况下具有有限个局部极小值点，且所有极小值点的特征值互不相同。我们还开发了一种黎曼信赖域方法，该方法相比经典算法有显著改进，可处理规模大得多的矩阵，同时还开发了一种变体用于计算最接近的实正规矩阵。本文通过大量数值实验对上述内容进行了补充验证。
英文摘要
We propose an approach based on Riemannian optimization to compute a nearest normal matrix to a given one. The problem can be formulated as the minimization of a smooth function either on the manifold $U(n)$ of unitary matrices of size n or on the flag manifold $U (n)/U (1)^n$. The flag manifold is particularly suitable for theoretical analysis; we characterize the global maximum of the objective function and prove that, for generic inputs, its local minimizers are finitely many and isolated; in turn, this implies the original nearest normal matrix problem generically has finitely many local minimizers, all with distinct eigenvalues. We also develop a Riemannian trust-region method that improves substantially on classical algorithms and can handle considerably larger matrices, as well as a variant for computing the nearest real normal matrix. The paper is complemented by extensive numerical experiments.
