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 areσ_r^2||ΔQ||_F^2≤||ΔA||_F^2, 1/2||ΔH||_F^2≤||ΔA||_F^2 and ||Δ∑||_F^2≤||ΔA||_F^2,respectively,where∑=diag(σ_1,σ_2,...,σ_r,0,...,0) is the singular value matrix of A andσ_r denotes the smallest nonzero singular value.Here we present some new combined (asymptotic) perturbation boundsσ_r^2||ΔQ||_F^2+1/2||ΔH||_F^2≤||ΔA||_F^2 andσ_r^2||ΔQ||_F^2+||Δ∑||_F^2≤||ΔA||_F^2 which are optimal for each factor.Some corresponding absolute perturbation bounds are also given.
Wen LI~(1+) Wei-wei SUN~2 1 School of Mathematical Sciences,South China Normal University,Guangzhou 510631,China
