The Polar Decomposition - Properties, Applications And Algorithms

Abstract

In the paper we review the numerical methods for computing the polar decomposition of a matrix. Numerical tests comparing these methods are included. Moreover, the applications of the polar decomposition and the most important its properties are mentioned. 1 Introduction In recent years interests in the polar decomposition have increased. Many interesting papers have appeared on properties, applications and numerical methods for this decomposition. In the paper we review the most important results concerning this very useful tool. Also we present numerical experiments comparing several algorithms for computing it. The polar decomposition was introduced by Autonne [1] in 1902. A thorough discussion of the history of it is given in Horn and Johnson [29, Sect. 3.0]. Let A be an arbitrary complex matrix, A 2 C m\Thetan . A polar decomposition of A is a factorization A = UH; (1) where H 2 C n\Thetan is Hermitian positive semi-definite matrix, H = H H and x H Hx 0 for every x 2 C..