We study the inversion analog of the well-known Gauss algorithm for multiplying complex matrices. A simple version is $(A + iB)^{-1} = (A + BA^{-1}B)^{-1} - i A^{-1}B(A+BA^{-1} B)^{-1}$ when $A$ is invertible, which may be traced back to Frobenius but has received scant attention. We prove that it is optimal, requiring fewest matrix multiplications and inversions over the base field, and we extend it in three ways: (i) to any invertible $A + iB$ without requiring $A$ or $B$ be invertible; (ii) to any iterated quadratic extension fields, with $\mathbb{C}$ over $\mathbb{R}$ a special case; (iii) to Hermitian positive definite matrices $A + iB$ by exploiting symmetric positive definiteness of $A$ and $A + BA^{-1}B$. We call all such algorithms Frobenius inversions, which we will see do not follow from Sherman--Morrison--Woodbury type identities and cannot be extended to Moore--Penrose pseudoinverse. We show that a complex matrix with well-conditioned real and imaginary parts can be arbitrarily ill-conditioned, a situation tailor-made for Frobenius inversion. We prove that Frobenius inversion for complex matrices is faster than standard inversion by LU decomposition and Frobenius inversion for Hermitian positive definite matrices is faster than standard inversion by Cholesky decomposition. We provide extensive numerical experiments, applying Frobenius inversion to solve linear systems, evaluate matrix sign function, solve Sylvester equation, and compute polar decomposition, showing that Frobenius inversion can be more efficient than LU/Cholesky decomposition with negligible loss in accuracy. A side result is a generalization of Gauss multiplication to iterated quadratic extensions, which we show is intimately related to the Karatsuba algorithm for fast integer multiplication and multidimensional fast Fourier transform.

翻译：我们研究了已知的用于复矩阵乘法的Gauss算法的求逆类比。一个简单版本是当$A$可逆时，$(A + iB)^{-1} = (A + BA^{-1}B)^{-1} - i A^{-1}B(A+BA^{-1} B)^{-1}$，该公式可追溯至Frobenius但鲜有关注。我们证明该公式是最优的，在基域上需要最少的矩阵乘法和求逆次数，并将其推广至三个方面：(i) 任意可逆的$A + iB$，无需$A$或$B$可逆；(ii) 任意迭代二次扩张域，其中$\mathbb{C}$ over $\mathbb{R}$作为特例；(iii) Hermite正定矩阵$A + iB$，利用$A$和$A + BA^{-1}B$的对称正定性。我们将此类算法统称为Frobenius求逆，并指出它们并非源于Sherman--Morrison--Woodbury型恒等式，且无法推广至Moore--Penrose伪逆。我们证明，实部和虚部均良态的复矩阵可能具有任意病态性，这正是Frobenius求逆的适用场景。我们证明，对于复矩阵，Frobenius求逆比基于LU分解的标准求逆更快；对于Hermite正定矩阵，Frobenius求逆比基于Cholesky分解的标准求逆更快。我们通过大量数值实验，将Frobenius求逆应用于求解线性方程组、计算矩阵符号函数、求解Sylvester方程以及计算极分解，结果表明Frobenius求逆比LU/Cholesky分解更高效，且精度损失可忽略。一个附带结果是Gauss乘法在迭代二次扩张上的推广，我们发现该推广与Karatsuba快速整数乘法算法及多维快速傅里叶变换密切相关。