Variation of the Unitary Part of a Matrix
Rajendra Bhatia, Kalyan K. Mukherjea · SIAM Journal on Matrix Analysis and Applications · 1994
The derivative of the map that takes an invertible matrix A to the unitary factor U in the polar decomposition $A = UP$ is evaluated. The same is done for the map that takes A to the unitary factor Q in the QR decomposition $A = QR$. These results lead to perturbation bounds for these maps. Other applications of the method developed are discussed.