The Canonical Generalized Polar Decomposition

Higham, Nicholas J. and Mehl, Christian and Tisseur, Françoise (2010) The Canonical Generalized Polar Decomposition. SIAM Journal On Matrix Analysis and Applications, 31 (4). 2163 -2180. ISSN 0895-4798

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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 $(M,N)$-partial 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.

Item Type: Article
Uncontrolled Keywords: generalized polar decomposition, canonical polar decomposition, automorphism, selfadjoint matrix, bilinear form, sesquilinear form, scalar product, adjoint, orthosymmetric scalar product, partial isometry, pseudoinverse, matrix sign function, matrix square root, matrix iteration
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Nick Higham
Date Deposited: 05 Jul 2010
Last Modified: 20 Oct 2017 14:12

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