We investigate a geometric computational framework, called the "scaling-rotation framework", on Sym + (p), the set of p × p symmetric positive-definite (SPD) matrices. The purpose of our study is to lay geometric foundations for statistical analysis of SPD matrices, in situations in which eigenstructure is of fundamental importance, for example diffusion-tensor imaging (DTI). Eigen-decomposition, upon which the scaling-rotation framework is based, determines both a stratification of Sym + (p), defined by eigenvalue multiplicities, and fibers of the "eigencomposition" map SO(p) × Diag + (p) → Sym + (p). This leads to the notion of scaling-rotation distance [Jung et al. (2015)], a measure of the minimal amount of scaling and rotation needed to transform an SPD matrix, X, into another, Y, by a smooth curve in Sym + (p). Our main goal in this paper is the systematic characterization and analysis of minimal smooth scalingrotation (MSSR) curves, images in Sym + (p) of minimal-length geodesics connecting two fibers in the "upstairs" space SO(p)×Diag + (p). The length of such a geodesic connecting the fibers over X and Y is what we define to be the scaling-rotation distance from X to Y. For the important lowdimensional case p = 3 (the home of DTI), we find new explicit formulas for MSSR curves and for the scaling-rotation distance, and identify M(X, Y ) in all "nontrivial" cases. The quaternionic representation of SO(3) is used in these computations. We also provide closed-form expressions for scalingrotation distance and MSSR curves for the case p = 2.MSC 2010 subject classifications: Primary 53C99; secondary 53C15, 53C22, 51F25, 15A18.
Groisser et al./Eigenvalue stratification and MSSR curves(For any subgroup H ⊂ O(p), we write S(H) for H SO(p).) In this example, G J has two connected components, one in which det(R 1 ) = det(R 2 ) = 1 (the component G 0 J ), and one in which det(R 1 ) = det(R 2 ) = −1. For general J = {J 1 , . . . , J r }, the elements of G J have "interleaved blocks". Writing k i = |J i |, we have(2.4) and the identity component G 0 J is isomorphic to SO(k 1 ) × SO(k 2 ) · · · × SO(k r ). If the k i are non-decreasing then [J] = k 1 + · · · + k r . For concreteness we define(2.5)The groups G J are also partially-ordered. For J, K ∈ Part({1, . . . , p}),This partial-ordering will be reflected in the stratifications of Sym + (p) and M discussed in Section 2.7.Definition 2.4 For each D ∈ Diag(p), we define the stabilizer group of D,