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The set of covariance matrices equipped with the Bures-Wasserstein distanceis the orbit space of the smooth, proper and isometric action of the orthogonalgroup on the Euclidean space of square matrices. This construction induces anatural orbit stratification on covariance matrices, which is exactly thestratification by the rank. Thus, the strata are the manifolds of symmetricpositive semi-definite (PSD) matrices of fixed rank endowed with theBures-Wasserstein Riemannian metric. In this work, we study the geodesics ofthe Bures-Wasserstein distance. Firstly, we complete the literature ongeodesics in each stratum by clarifying the set of preimages of the exponentialmap and by specifying the injection domain. We also give explicit formulae ofthe horizontal lift, the exponential map and the Riemannian logarithms thatwere kept implicit in previous works. Secondly, we give the expression of allthe minimizing geodesic segments joining two covariance matrices of any rank.More precisely, we show that the set of all minimizing geodesics between twocovariance matrices $\Sigma$ and $\Lambda$ is parametrized by the closed unitball of $\mathbb{R}^{(k-r)\times(l-r)}$ for the spectral norm, where $k, l, r$are the respective ranks of $\Sigma$, $\Lambda$, $\Sigma$$\Lambda$. Inparticular, the minimizing geodesic is unique if and only if $r = \min(k, l)$.Otherwise, there are infinitely many.

Bures-Wasserstein minimizing geodesics between covariance matrices of different ranks