Perturbation Bounds for the Polar Decomposition

Let M n (F) denote the space of matrices over the eld F. Given A2 M n (F) deene jAj (A A) 1=2 and U(A) AjAj ?1 assuming A is nonsingular. Let 1 (A) 2 (A) n (A) 0 denote the ordered singular values of A. We obtain majorization results relating the singular values of U(A + A) ? U(A) and those of A and A. In particular we show that if A; A2 M n (R) and 1 ((A) < n (A) then for any unitarily invariant norm k k, kU(A + A) ? U(A)k 22 n?1 (A) + n (A)] ?1 kAk. We obtain similar results for matrices with complex entries. We also consider the unitary Procrustes problem: minfkA ? UBk : U2 M n (C); U U = Ig where A; B2 M n (C) and a unitarily invariant norm k k are given. It was conjectured that if U is unitary and U BA is positive semideenite then U must be a solution to the unitary Procrustes problem for all unitarily invariant norms. We show that the conjecture is false.