Combined perturbation bounds : II. Polar decompositions

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)22_Publication in policy or professional journal

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Original languageEnglish
Pages (from-to)1339-1346
Journal / PublicationScience in China, Series A: Mathematics
Issue number9
Publication statusPublished - Sept 2007


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. © 2007 Science in China Press.

Research Area(s)

  • Perturbation, Polar decomposition, Singular value