AI 中文总结
该研究将经典优序关系的对齐序放宽为偏序，借助基变换矩阵的LU近似强化优序，得到多个经典优序关系的细化版本，并将其推广到张量积和等场景。
AI 中文摘要
矩阵和或乘积的若干经典优序关系涉及完全对齐的特征值或奇异值的优序向量。通过将对齐的序放宽为偏序，我们证明，若基变换矩阵关于该偏序允许LU近似，则优序可被强化。由此，我们得到Ky Fan优序关系、Horn对数优序关系及von Neumann迹不等式的细化版本。作为应用，我们给出任意多个张量因子的可分Ky Fan优序关系的简短证明，并将其推广到任意矩阵张量积的和。进一步应用涉及（反）对称幂和的优序关系及Kronecker和乘积的优序关系。
英文摘要
Several classical majorization relations for sums or products of matrices involve a majorizing vector of perfectly aligned eigenvalues or singular values. By relaxing the order of alignment to a partial order, we show that the majorization can be strengthened, provided the change-of-basis matrices admit an LU-approximation with respect to this partial order. In this way, we obtain refined versions of Ky Fan's majorization relations, Horn's log-majorization relation, and von Neumann's trace inequality. As an application, we give a short proof of the separable Ky Fan majorization relation for an arbitrary number of tensor factors and extend it to a sum of tensor products of arbitrary matrices. Further applications concern majorization relations for sums of (anti-)symmetric powers and for products of Kronecker sums.
Comments13 pages
