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 areσr2||ΔQ||F2≤||ΔA||F2, 1/2||ΔH||F2≤||ΔA||F2 and ||Δ∑||F2≤||ΔA||F2,respectively,where∑=diag（σ1,σ2,...,σr,0,...,0） is the singular value matrix of A andσr denotes the smallest nonzero singular value.Here we present some new combined （asymptotic） perturbation boundsσr2||ΔQ||F2+1/2||ΔH||F2≤||ΔA||F2 andσr2||ΔQ||F2+||Δ∑||F2≤||ΔA||F2 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
