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：我们分析了Frobenius首先提出的复杂的矩阵转换算法,但基本上被遗忘了:$(A+ iB)%-1} = (A+ BA ⁇ -1}B) =-1} - i A*-1} B(A+B) B(A+B ⁇ -1}B) ⁇ -1} $是不可逆的,而美元(A+iB) = B-1} A(AB}-1}A+B) *-1} - (i(AB ⁇ -1}A+B) +B) *-1} 美元,当美元是不可逆的时。这可以被视为(A+BA+B ⁇ -1}) = (A+BA+B ⁇ -1}B) * * *-1} - -1} - i A 美元是不可逆的,当美元是(A+BA+B} (A+BA+B}-1}-1}B}B} } } } = = = $ (A+ + iBB) = 1} = = = = = (A+ +Bax = 1} = = = = = = (A+Ba) = = = = = = = = = = = = = (A+B= =) = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = (A= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =