计算极分解的Frechet导数
Computing the Fréchet Derivative of the Polar Decomposition
AI总结：
本文提出迭代方法计算极分解中矩阵U的Frechet导数，利用新身份将导数与矩阵符号函数关联，适用于正方形和矩形矩阵。
摘要：
我们推导出用于计算极分解映射的迭代方法，该映射将满秩矩阵A映射到其极分解A=UH中的因子U，其中U有正交列，H为Hermitian正定矩阵。这些方法适用于正方形矩阵以及行数多于列数的矩形矩阵。我们的推导依赖于一个新身份，该身份将极分解的Frechet导数与应用于特定块矩阵X的矩阵符号函数sign(X)=X(X²)^{-1/2}关联。
英文摘要：
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 $\mathrm{sign}(X) = X (X^2)^{-1/2}$ applied to a certain block matrix $X$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
