  We investigate a geometric computational framework, called the
"scaling-rotation framework", on ${\rm Sym}^+(p)$, the set of $p \times p$
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 ${\rm
Sym}^+(p)$, defined by eigenvalue multiplicities, and fibers of the
"eigen-composition" map $SO(p)\times{\rm Diag}^+(p)\to{\rm Sym}^+(p)$. 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 ${\rm Sym}^+(p)$. Our
main goal in this paper is the systematic characterization and analysis of
minimal smooth scaling-rotation (MSSR) curves, images in ${\rm Sym}^+(p)$ of
minimal-length geodesics connecting two fibers in the "upstairs" space
$SO(p)\times{\rm Diag}^+(p)$. 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 low-dimensional case $p = 3$ (the home of
DTI), we find new explicit formulas for MSSR curves and for the
scaling-rotation distance, and identify ${\cal M}(X,Y)$ in all "nontrivial"
cases. The quaternionic representation of $SO(3)$ is used in these
computations. We also provide closed-form expressions for scaling-rotation
distance and MSSR curves for the case $p = 2$.
