Dynamical Methods for Polar Decomposition and Inversion of Matrices

Getz, N. H. and J. E. Marsden

Linear Algebra and its Appl., 258, 311-343

Abstract:

We show how one may obtain polar decomposition as well as inversion of fixed and time-varying matrices using a class of nonlinear continuous-time dynamical systems. First we construct a dynamical system that causes an initial approximation of the inverse of a time-varying matrix to flow exponentially toward the true time-varying inverse. Using a time-parameterized homotopy from the identity matrix to a fixed matrix with unknown inverse, and applying our result on the inversion of time-varying matrices, we show how any positive definite fixed 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 definite symmetric part of the polar decomposition. As a byproduct, this method gives another method of inverting time-varying matrices. Finially, using homotopy again, we show how dynamic polar decomposition may be applied to fixed matrices with the added benefit that this allows us to dynamically invert any fixed matrix by a prescribed time.