Uniqueness questions in a scaling-rotation geometry on the space of symmetric positive-definite matrices

Jung et al. (2015) introduced a geometric structure on Sym^+(p), the set of p × p symmetric positive-definite matrices, based on eigen-decomposition. Eigenstructure determines both a stratification of Sym^+(p), defined by eigenvalue multiplicities, and fibers of the "eigen-composition" map F:M(p):=SO(p)× Diag^+(p)→ 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∈ 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 Sym^+(p) of minimal-length geodesics connecting two fibers. In this paper we study the geometry of the triple (M(p),F, 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 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( R^p), with m even, that we answer for p≤ 4 and p≥ 11. Our method of proof also yields an interesting half-angle formula concerning principal angles between subspaces of 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. (2017). Addressing the uniqueness-related questions requires a thorough understanding of the fiber structure of M(p), which we also provide.