This paper is a self-contained exposition of the geometry of symmetric positive-definite real \(n\times n\) matrices \({\text {SPD}}(n)\) , including necessary and sufficent conditions for a submanifold \(\mathcal {N} \subset {\text {SPD}}(n)\) to be totally geodesic for the affine-invariant Riemannian metric. A non-linear projection \(x\mapsto \pi (x)\) on a totally geodesic submanifold is defined. This projection has the minimizing property with respect to the Riemannian metric: it maps an arbitrary point \(x \in {\text {SPD}}(n)\) to the unique closest element \(\pi (x)\) in the totally geodesic submanifold for the distance defined by the affine-invariant Riemannian metric. Decompositions of the space \({\text {SPD}}(n)\) follow, as well as variants of the polar decomposition of non-singular matrices known as Mostow’s decompositions. Applications to decompositions of covariant matrices are mentioned.
