Combined perturbation bounds: II. Polar decompositions

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.