The Perturbation Bounds For The Generalized Polar Decomposition And Weighted Polar Decomposition

The generalized polar decomposition and weighted polar decomposition play an important role in many fields of numerical analysis, matrix approximation and so on, especially on the matrix theory research. Moreover, many researchers have already studied and presented a lot of classical perturbation results of the perturbation problems for generalized polar decomposition and weighted polar decomposition. Based on this, we further study the problem.In this paper, we study the perturbation of the generalized polar decomposition and weighted polar decomposition under the same rank. Firstly, some new pertur-bation bounds of positive (semi)definite polar factor and (sub)unitary polar factor for (generalized) polar decomposition under the general unitarily invariant norm and the spectral norm are presented; Secondly, according to the connection between generalized polar decomposition and weighted polar decomposition, we directly give out the perturbation bounds for weighted polar decomposition without proof which improve the known results. Finally, we further consider the multiplicative pertur-bation of the weighted polar decomposition, and present some new perturbation bounds for the generalized nonnegative and weighted unitary polar factors under the different weighted norms which improve the existing perturbation bounds.