PERTURBATION BOUNDS FOR THE POLAR DECOMPOSITION

Let M(n)(F) denote the space of matrices over the field F. Given A is-an-element-of M(n)(F) define Absolute value of A = (A*A)1/2 and U(A) = A Absolute value of A-1 assuming A is nonsingular. Let sigma1(A) greater-than-or-equal-to sigma2(A) greater-than-or-equal-to ... greater-than-or-equal-to sigma(n)(A) greater-than-or-equal-to 0 denote the ordered singular values of A. Majorization results are obtained relating the singular values of U(A + DELTAA) - U(A) and those of A and DELTAA. In particular, it is shown that if A, DELTAA is-an-element-of M(n)(R) and sigma1(DELTAA) < sigma(n)(A), then for any unitarily invariant norm \\.\\, \\U(A + DELTAA) - U(A)\\ less-than-or-equal-to 2[sigma(n-1) (A) + sigma(n)(A)]-1 \\DELTAA\\. Similar results are obtained for matrices with complex entries. Also considered is the unitary Procrustes problem: min{\\A - UB\\ : U is-an-element-of M(n)(C), U*U = I} where A, B is-an-element-of M(n)(C), and a unitarily invariant norm \\.\\ are given. It was conjectured that if U is unitary and U*BA* is positive semidefinite then U must be a solution to the unitary Procrustes problem for all unitarily invariant norms. The conjecture is shown to be false.