@article{c948646cf14a4b3ead5fd7f3d468dfc1, title = "Combined perturbation bounds: II. Polar decompositions", abstract = "In this paper, we study the perturbation bounds for the polar decomposition A = QH where Q is unitary and H is Hermitian. The optimal (asymptotic) bounds obtained in previous works for the unitary factor, the Hermitian factor and singular values of A are σ r 2 ΔQ F 2 Δ F 2 , 1/2ΔH F 2 ΔA F 2 and Δ∑ F 2 ΔA F 2 , respectively, where ∑ = diag(σ 1, σ 2, σ r , 0 ) is the singular value matrix of A and σ r denotes the smallest nonzero singular value. Here we present some new combined (asymptotic) perturbation bounds σ r 2 ΔQ F 2 +1/2 ΔH F 2 ΔA F 2 and σ r 2 Delta;Q F 2 + Δ∑ F 2 ΔA F 2 which are optimal for each factor. Some corresponding absolute perturbation bounds are also given. {\textcopyright} 2007 Science in China Press.", keywords = "Perturbation, Polar decomposition, Singular value", author = "Wen Li and Wei-Wei Sun", year = "2007", month = sep, doi = "10.1007/s11425-007-0099-z", language = "English", volume = "50", pages = "1339--1346", journal = "Science in China, Series A: Mathematics, Physics, Astronomy", issn = "1006-9283", publisher = "Science in China Press", number = "9", }