发表机构
University of Waterloo（滑铁卢大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明正交投影矩阵的主余子式决定其像的逐坐标绝对值，并通过拟阵连通性分层格拉斯曼流形，给出各层结构定理。
AI 中文摘要
我们证明，一个正交投影矩阵的主余子式决定其像的逐坐标绝对值。更精确地说，若$W_1,W_2\subset \mathbb{C}^n$是子空间，其正交投影矩阵的所有阶对应主余子式相等，则$|W_1|=|W_2|$，其中$|W|$是$W$在逐坐标绝对值映射下的像。证明根据相关拟阵的连通性分为不同情形。我们提供了格拉斯曼流形按连通性水平的分层，并给出了每一层元素的结构定理。
英文摘要
We prove that the principal minors of an orthogonal projection matrix determine the coordinatewise absolute value of its image. More precisely, if $W_1,W_2\subset \mathbb{C}^n$ are subspaces whose orthogonal projection matrices have equal corresponding principal minors of all orders, then $|W_1|=|W_2|$, where $|W|$ is the image of $W$ under the coordinatewise absolute value map. The proof is divided into different cases related to the connectivity of the associated matroids. We provide a stratification of the Grassmannian in terms of the connectivity levels and give a structure theorem for the elements of each stratum.
