Jung et al. [7] introduced a geometric structure on ${\mathrm{Sym}}^{+}(p)$, the set of $p\times p$ symmetric positive-definite matrices, based on eigen-decomposition. Eigenstructure determines both a stratification of ${\mathrm{Sym}}^{+}(p)$, defined by eigenvalue multiplicities, and fibers of the “eigen-composition” map $F:M(p):=SO(p)\times {\mathrm{Diag}}^{+}(p)\to {\mathrm{Sym}}^{+}(p)$. When $M(p)$ is equipped with a suitable Riemannian metric, the fiber structure leads to notions of scaling-rotation distance between $X,Y\in {\mathrm{Sym}}^{+}(p)$, the distance in $M(p)$ between fibers $F^{-1}(X)$ and $F^{-1}(Y)$, and minimal smooth scaling-rotation (MSSR) curves, images in ${\mathrm{Sym}}^{+}(p)$ of minimal-length geodesics connecting two fibers. In this paper we study the geometry of the triple $(M(p),F,{\mathrm{Sym}}^{+}(p))$, focusing on some basic questions: For which $X,Y$ is there a unique MSSR curve from X to Y? More generally, what is the set $\mathcal{M}(X,Y)$ of MSSR curves from X to Y? This set is influenced by two potential types of non-uniqueness. We translate the question of whether the second type can occur into a question about the geometry of Grassmannians $G_m({\mathbf{R}}^p)$, with m even, that we answer for $p\le 4$ and $p\ge 11$. Our method of proof also yields an interesting half-angle formula concerning principal angles between subspaces of ${\mathbf{R}}^p$ whose dimensions may or may not be equal. The general-p results concerning MSSR curves and scaling-rotation distance that we establish here underpin the explicit $p=3$ results in Groisser et al. [5]. Addressing the uniqueness-related questions requires a thorough understanding of the fiber structure of $M(p)$, which we also provide.

荣格等人。[7] 介绍了一种几何结构${\mathrm{符号}}^{+}(磷)$，集合 $磷\times 磷$基于特征分解的对称正定矩阵。特征结构决定了${\mathrm{符号}}^{+}(磷)$，由特征值多重性和“特征组合”映射的纤维定义 $F：米(磷)：=秒哦(磷)\times {\mathrm{诊断}}^{+}(磷)\to {\mathrm{符号}}^{+}(磷)$. 什么时候$米(磷)$配备了合适的黎曼度量，纤维结构导致之间的缩放 - 旋转距离的概念$X,是\in {\mathrm{符号}}^{+}(磷)$，距离在 $米(磷)$ 纤维之间 $F^{-1}(X)$ 和 $F^{-1}(是)$，以及最小平滑缩放-旋转 (MSSR) 曲线，图像${\mathrm{符号}}^{+}(磷)$连接两条光纤的最小长度测地线。在本文中，我们研究了三元组的几何$(米(磷),F,{\mathrm{符号}}^{+}(磷))$，关注一些基本问题：对于哪些 $X,是$从X到Y是否有唯一的MSSR 曲线？更一般地，什么是集合$\mathcal{米}(X,是)$从X到Y的 MSSR 曲线？该集合受到两种潜在类型的非唯一性的影响。我们将第二类能否出现的问题转化为关于格拉斯曼几何学的问题$G_{米}({\mathbf{电阻}}^{磷})$与米甚至，我们回答$磷\le 4$ 和 $磷\ge 11$. 我们的证明方法还产生了一个有趣的半角公式，涉及子空间之间的主角${\mathbf{电阻}}^{磷}$其尺寸可能相等，也可能不相等。在常规- p关于MSSR曲线和缩放，旋转距离，我们在这里建立巩固明确的结果$磷=3$Groisser 等人的结果。[5]。解决与唯一性相关的问题需要彻底了解$米(磷)$，我们也提供。