让矩阵更优秀：极值和奇异值分解的几何与拓扑
Making matrices better: Geometry and topology of polar and singular value decomposition
AI总结：
从几何与拓扑角度研究固定尺寸矩阵空间，探讨不同秩家族及其关联，分析矩阵的最近正交邻近和最近奇异邻近，以及极值和奇异值分解及其应用。
摘要：
本文旨在从几何和拓扑的角度研究给定尺寸矩阵的空间，重点探讨各种秩家族及其相互关联。特别关注给定矩阵的最近正交邻近和最近奇异邻近，这两者在矩阵分解中起核心作用。在此视觉背景下，研究极值和奇异值分解及其某些应用。
英文摘要：
Our goal here is to see the space of matrices of a given size from a geometric and topological perspective, with emphasis on the families of various ranks and how they fit together. We pay special attention to the nearest orthogonal neighbor and nearest singular neighbor of a given matrix, both of which play central roles in matrix decompositions, and then against this visual backdrop examine the polar and singular value decompositions and some of their applications.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
