Computing the Fr\'echet Derivative of the Polar Decomposition

TL;DR

This paper develops iterative algorithms to compute the Fréchet derivative of the polar decomposition map, applicable to both square and rectangular matrices, using a novel identity involving the matrix sign function.

Contribution

It introduces new iterative methods for the Fréchet derivative of the polar decomposition, based on a novel identity linking it to the matrix sign function.

Findings

Derived iterative methods for the Fréchet derivative

Applicable to square and rectangular matrices

Utilizes a new identity involving the matrix sign function

Abstract

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 $X$.