  We show how the Riemannian distance on $\mathbb{S}^n_{++}$, the cone of
$n\times n$ real symmetric or complex Hermitian positive definite matrices, may
be used to naturally define a distance between two such matrices of different
dimensions. Given that $\mathbb{S}^n_{++}$ also parameterizes $n$-dimensional
ellipsoids, and inner products on $\mathbb{R}^n$, $n \times n$ covariance
matrices of nondegenerate probability distributions, this gives us a natural
way to define a geometric distance between a pair of such objects of different
dimensions.
