We derive iterative methods for computing the Fr\'{ e}chet 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 Fr\'{ e}chet derivative of the polar decomposition to the matrix sign function $\mathrm{ sign}(X) = X (X^2)^{ -1/2}$ applied to a certain block matrix $X$.
Categories: math.NA