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### Open Access Journals

The space of probability densities is an infinite-dimensional Riemannian manifold, with Riemannian metrics in two flavors: Wasserstein and Fisher-Rao. The former is pivotal in optimal mass transport (OMT), whereas the latter occurs in information geometry--the differential geometric approach to statistics. The Riemannian structures restrict to the submanifold of multivariate Gaussian distributions, where they induce Riemannian metrics on the space of covariance matrices.

Here we give a systematic description of classical matrix decompositions (or factorizations) in terms of Riemannian geometry and compatible principal bundle structures. Both Wasserstein and Fisher-Rao geometries are discussed. The link to matrices is obtained by considering OMT and information geometry in the category of linear transformations and multivariate Gaussian distributions. This way, OMT is directly related to the polar decomposition of matrices, whereas information geometry is directly related to the $QR$, Cholesky, spectral, and singular value decompositions. We also give a coherent description of gradient flow equations for the various decompositions; most flows are illustrated in numerical examples.

The paper is a combination of previously known and original results. As a survey it covers the Riemannian geometry of OMT and polar decompositions (smooth and linear category), entropy gradient flows, and the Fisher-Rao metric and its geodesics on the statistical manifold of multivariate Gaussian distributions. The original contributions include new gradient flows associated with various matrix decompositions, new geometric interpretations of previously studied isospectral flows, and a new proof of the polar decomposition of matrices based an entropy gradient flow.

*contact oscillator*, which has been used as a test problem for numerical methods for nonholonomic mechanics. The systems under study constitute simple models for continuously variable transmission gearboxes.

The main result is that each system in the family is integrable reversible with respect to the canonical reversibility map on the cotangent bundle. By using reversible Kolmogorov--Arnold--Moser theory, we then establish preservation of invariant tori for reversible perturbations. This result explains previous numerical observations, that some discretisations of the contact oscillator have favourable structure preserving properties.

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