On the perturbation bound in unitarily invariant norms for subunitary polar factors

Let Crm×n be the set of m×nm×n complex matrices with rank r  , and let A∈Crm×n and A∼=A+E∈Crm×n have the generalized polar decompositionsA=QHandA∼=Q∼H∼.In this article, a new perturbation bound for subunitary polar factors in any unitarily invariant norm is given by∥Q∼-Q∥⩽3σr+σ˜r∥E∥,where σr,σ∼r are the smallest positive singular values of A   and A∼, respectively, which improves some existing perturbation bounds.