The canonical generalized polar decomposition

The polar decomposition of a square matrix has been generalized by several authors to scalar products on ℝn or ℂn given by a bilinear or sesquilinear form. Previous work has focused mainly on the case of square matrices, sometimes with the assumption of a Hermitian scalar product. We introduce the canonical generalized polar decomposition A = WS, defined for general m × n matrices A, where W is a partial (M,N)-isometry and S is N-selfadjoint with nonzero eigenvalues lying in the open right half-plane, and the nonsingular matrices M and N define scalar products on ℂm and ℂn, respectively. We derive conditions under which a unique decomposition exists and show how to compute the decomposition by matrix iterations. Our treatment derives and exploits key properties of partial (M,N)-isometries and orthosymmetric pairs of scalar products, and also employs an appropriate generalized Moore-Penrose pseudoinverse. We relate commutativity of the factors in the canonical generalized polar decomposition to an appropriate definition of normality. We also consider a related generalized polar decomposition A = WS, defined only for square matrices A and in which W is an automorphism; we analyze its existence and the uniqueness of the selfadjoint factor when A is singular. Copyright © 2010 Society for Industrial and Applied Mathematics.