Dynamical Methods for Polar Decomposition and Inversion of Matrices

We show how one may obtain polar decomposition as well as inversion of xed and time-varying matrices using a class of nonlinear continuous-time dynamical systems. First we construct a dynamic system that causes an initial approximation of the inverse of a time-varying matrix to ow exponentially toward the true time-varying inverse. Using a time-parameterized homotopy from the identity matrix to a xed matrix with unknown inverse, and applying our result on the inversion of time-varying matrices, we show how any positive de nite xed matrix may be dynamically inverted by a prescribed time without an initial guess at the inverse. We then construct a dynamical system that solves for the polar decomposition factors of a time-varying matrix given an initial approximation for the inverse of the positive de nite symmetric part of the polar decomposition. As a byproduct, this method gives another method of inverting time-varying matrices. Finally, using homotopy again, we show how dynamic polar decomposition may be applied to xed matrices with the added bene t that this allows us to dynamically invert any xed matrix by a prescribed time.