Higham, Nicholas J. and Mehl, Christian and Tisseur, Françoise (2009) The Canonical Generalized Polar Decomposition. [MIMS Preprint]
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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 halfplane, 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 MoorePenrose 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:  MIMS Preprint 

Additional Information:  To appear in the SIAM Journal On Matrix Analysis and Applications 
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:  18 Mar 2010 
Last Modified:  08 Nov 2017 18:18 
URI:  https://eprints.maths.manchester.ac.uk/id/eprint/1427 
Available Versions of this Item

The Canonical Generalized Polar Decomposition. (deposited 14 Jul 2009)
 The Canonical Generalized Polar Decomposition. (deposited 18 Mar 2010) [Currently Displayed]
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