Volume: 41 • Number: 1 • Pages: 171-198

This paper explores the well-known identification of the manifold of rank p positivesemidefinitematricesofsizenwiththequotientofthesetoffull-rankn-by-pmatricesbytheorthogonal group in dimension p. The induced metric corresponds to the Wasserstein metric between centered degenerate Gaussian distributions, and is a generalization of the Bures–Wasserstein metric on the manifoldofpositive-definitematrices. WecomputetheRiemannianlogarithm,andshowthatthelocal injectivity radiusat anymatrix C isthesquareroot ofthe pth largesteigenvalue of C. Asa result, the globalinjectivityradiusonthismanifoldiszero. Finally,thispaperalsocontainsadetaileddescription of this geometry, recovering previously known expressions by applying the standard machinery of Riemannian submersions.