We analyze a complex matrix inversion algorithm first proposed by Frobenius, but largely forgotten: $(A + iB)^{-1} = (A + BA^{-1}B)^{-1} - i A^{-1}B(A+BA^{-1} B)^{-1}$ when $A$ is invertible and $(A + iB)^{-1} = B^{-1}A(AB^{-1}A + B)^{-1} - i (AB^{-1}A + B)^{-1}$ when $B$ is invertible. This may be viewed as an inversion analogue of the aforementioned Gauss multiplication. We proved that Frobenius inversion is optimal -- it uses the least number of real matrix multiplications and inversions among all complex matrix inversion algorithms. We also showed that Frobenius inversion runs faster than the standard method based on LU decomposition if and only if the ratio of the running time for real matrix inversion to that for real matrix multiplication is greater than $5/4$. We corroborate this theoretical result with extensive numerical experiments, applying Frobenius inversion to evaluate matrix sign function, solve Sylvester equation, and compute polar decomposition, concluding that for these problems, Frobenius inversion is more efficient than LU decomposition with nearly no loss in accuracy.

翻译：本文分析了一种由Frobenius首次提出但长期被忽视的复数矩阵求逆算法：当$A$可逆时，有$(A + iB)^{-1} = (A + BA^{-1}B)^{-1} - i A^{-1}B(A+BA^{-1} B)^{-1}$；当$B$可逆时，有$(A + iB)^{-1} = B^{-1}A(AB^{-1}A + B)^{-1} - i (AB^{-1}A + B)^{-1}$。该算法可视为前述高斯乘法在求逆运算中的对应形式。我们证明了Frobenius求逆法具有最优性——在所有复数矩阵求逆算法中，其使用的实矩阵乘法与求逆运算次数最少。同时研究表明：当实矩阵求逆与实矩阵乘法的运行时间之比大于$5/4$时，Frobenius求逆法比基于LU分解的标准方法运行更快。通过大规模数值实验验证了该理论结果，将Frobenius求逆法应用于矩阵符号函数求值、Sylvester方程求解及极分解计算，结果表明在这些问题中，Frobenius求逆法相较于LU分解具有更高效率，且精度几乎无损。