  We study extensions of Fr\'{e}chet means for random objects in the space
${\rm Sym}^+(p)$ of $p \times p$ symmetric positive-definite matrices using the
scaling-rotation geometric framework introduced by Jung et al. [\textit{SIAM J.
Matrix. Anal. Appl.} \textbf{36} (2015) 1180-1201]. The scaling-rotation
framework is designed to enjoy a clearer interpretation of the changes in
random ellipsoids in terms of scaling and rotation. In this work, we formally
define the \emph{scaling-rotation (SR) mean set} to be the set of Fr\'{e}chet
means in ${\rm Sym}^+(p)$ with respect to the scaling-rotation distance. Since
computing such means requires a difficult optimization, we also define the
\emph{partial scaling-rotation (PSR) mean set} lying on the space of
eigen-decompositions as a proxy for the SR mean set. The PSR mean set is easier
to compute and its projection to ${\rm Sym}^+(p)$ often coincides with SR mean
set. Minimal conditions are required to ensure that the mean sets are
non-empty. Because eigen-decompositions are never unique, neither are PSR
means, but we give sufficient conditions for the sample PSR mean to be unique
up to the action of a certain finite group. We also establish strong
consistency of the sample PSR means as estimators of the population PSR mean
set, and a central limit theorem. In an application to multivariate
tensor-based morphometry, we demonstrate that a two-group test using the
proposed PSR means can have greater power than the two-group test using the
usual affine-invariant geometric framework for symmetric positive-definite
matrices.
