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 σr2‖ΔQ‖F2 ⩽ ‖ΔA‖F2, 1/2‖ΔH‖F2 ⩽ ‖ΔA‖F2 and ‖ΔΣ‖F2 ⩽ ‖ΔA‖F2, respectively, where Σ = diag(σ1, σ2, …, σr, …, 0 ) is the singular value matrix of A and σr denotes the smallest nonzero singular value. Here we present some new combined (asymptotic) perturbation bounds σr2‖ΔQ‖F2+1/2‖ΔH‖F2 ⩽ ‖ΔA‖F2 and σr2‖ΔQ‖F2+‖ΔΣ‖F2 ⩽ ‖ΔA‖F2 which are optimal for each factor. Some corresponding absolute perturbation bounds are also given.