We show how the Riemannian distance on n++, the cone of n×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 n++ also parameterizes n-dimensional ellipsoids, and inner products on ℝn, n×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.