  We investigate a matrix dynamical system related to optimal mass transport in
the linear category, namely, the problem of finding an optimal invertible
matrix by which two covariance matrices are congruent. We first review the
differential geometric structure of the problem in terms of a principal fiber
bundle. The dynamical system is a gradient flow restricted to the fibers of the
bundle. We prove global existence of solutions to the flow, with convergence to
the polar decomposition of the matrix given as initial data. The convergence is
illustrated in a numerical example.
