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

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