DYNAMICAL METHODS FOR POLAR DECOMPOSITION AND INVERSION OF MATRICES

We show how to obtain polar decomposition as well as inversion of fixe d 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 no w exponentially toward the true time-varying inverse. Using a time-par ametrized homotopy from the identity matrix to a fixed matrix with unk nown inverse, and applying our result on the inversion of time-varying matrices, we show how any positive definite fixed matrix may be dynam ically inverted by a prescribed time without an initial guess at the i nverse. We then construct a dynamical system that solves for the polar decomposition factors of a time-varying matrix given an initial appro ximation for the inverse of the positive definite symmetric part of th e polar decomposition. As a by-product, this method gives another meth od of inverting time-varying matrices. Finally, using homotopy again, we show how dynamic polar decomposition may be applied to fixed matric es with the added benefit that this allows us to dynamically invert an y fixed matrix by a prescribed time. (C) Elsevier Science Inc., 1997.