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

Abstract

Jung et al. [7] 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 Gm(Rp), 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 Rp 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.