We derive iterative methods for computing the Fréchet derivative of the map which 6 sends a full-rank matrix A to the factor U in its polar decomposition A = UH, where U has 7 orthonormal columns and H is Hermitian positive definite. The methods apply to square matrices 8 as well as rectangular matrices having more rows than columns. Our derivation relies on a novel 9 identity that relates the ...

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 invaria...

Let A = UP be a polar decomposition of an n×n complex matrix A. Then for every unitarily invariant norm ||| · |||, it is shown that ||| |UP − PU |||| ≤ |||A∗A−AA∗||| ≤ ‖UP + PU‖ |||UP − PU |||, where ‖·‖ denotes the operator norm. This is a quantitative version of the wellknown result that A is normal if and only if UP = PU . Related inequalities involving self-commutators are also obtained.

Three different kinds of condition numbers: normwise, mixed and componentwise, are discussed for the canonical generalized polar decomposition (CGPD) of real matrices. The technique used herein is different from the ones in previous literatures of the polar decomposition. With some modifications of the definition of the componentwise condition number, its application scope is extended. Explicit...