Geometric distance between positive definite matrices of different dimensions

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.