Iterative Computation of the Fréchet Derivative of the Polar Decomposition

We derive iterative methods for computing the Fréchet derivative of the map which sends a full-rank matrix A to the factor U in its polar decomposition A = UH, where U has orthonormal columns and H is Hermitian positive definite. The methods apply to square matrices as well as rectangular matrices having more rows than columns. Our derivation relies on a novel identity that relates the Fréchet derivative of the polar decomposition to the matrix sign function sign(X) = X(X2)−1/2 applied to a certain block matrix X.