Matrix analysis
Computing the polar decomposition with applications
SIAM Journal on Scientific and Statistical Computing
On scaling Newton's method for polar decomposition and the matrix sign function
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Functions Preserving Matrix Groups and Iterations for the Matrix Square Root
SIAM Journal on Matrix Analysis and Applications
Structured Factorizations in Scalar Product Spaces
SIAM Journal on Matrix Analysis and Applications
Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics)
Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics)
A New Scaling for Newton's Iteration for the Polar Decomposition and its Backward Stability
SIAM Journal on Matrix Analysis and Applications
Weighted Polar Decomposition and WGL Partial Ordering of Rectangular Complex Matrices
SIAM Journal on Matrix Analysis and Applications
Entanglement capabilities of the spin representation of (3+1)D-conformal transformations
Quantum Information & Computation
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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 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.