A Refined Polar Decomposition: A=UPD

A reenement of the polar decomposition of a nonsingular matrix A is considered. Here A is written as a product of unitary U ; Hermitian and positive deenite P which has unit diagonal, and diagonal positive D : It is shown that such a decomposion exists and is unique. Rectangular and singular cases are also considered. Then a simple xed point iteration using SVD is given to compute this decomposition. Also implementation of the Newton's method is discussed The reened polar decomposition can be used to parameterize the orbit of a matrix with distinct eigenvalues.