Authors’ abstract: The polar decomposition of a square matrix has been generalized by several authors to scalar products on \(\mathbb R^n\) or \(\mathbb C^n\) 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 \(m\times n\) 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 \(\mathbb C^m\) and \(\mathbb C^n\), 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.