## 2009.52: The Canonical Generalized Polar Decomposition

2009.52:
Nicholas J. Higham, Christian Mehl and Françoise Tisseur
(2009)
*The Canonical Generalized Polar Decomposition.*

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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 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: | MIMS Preprint |
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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 2000 > 15 Linear and multilinear algebra; matrix theory MSC 2000 > 65 Numerical analysis |

MIMS number: | 2009.52 |

Deposited By: | Nick Higham |

Deposited On: | 14 July 2009 |

### Available Versions of this Item

- The Canonical Generalized Polar Decomposition (deposited 05 July 2010)
- The Canonical Generalized Polar Decomposition (deposited 18 March 2010)
- The Canonical Generalized Polar Decomposition (deposited 14 July 2009)
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