我们研究了与著名的复矩阵乘法高斯算法相对应的求逆方法。当矩阵A可逆时，该方法的简单形式为$(A + iB)^{-1} = (A + BA^{-1}B)^{-1} - i A^{-1}B(A+BA^{-1} B)^{-1}$，这一公式可追溯至Frobenius，但此前受到的关注极少。我们证明该方法是最优的，在基域上所需的矩阵乘法与求逆次数最少，并从三个方向对其进行了扩展：（i）无需A或B可逆，适用于任意可逆的$A + iB$矩阵；（ii）适用于任意迭代二次扩域，其中复数域$\u2102$在实数域$\u211d$上的扩张是特例；（iii）利用A与$A + BA^{-1}B$的对称正定性，将其推广到厄米特正定矩阵$A + iB$的求逆场景。我们将这类算法统称为Frobenius求逆（Frobenius inversions），研究表明这类算法并非源自Sherman–Morrison–Woodbury类恒等式，也无法推广到Moore–Penrose伪逆的计算。我们证明了实部和虚部条件数良好的复矩阵可能存在任意差的条件数，这种场景恰好适合使用Frobenius求逆。我们还证明，复矩阵的Frobenius求逆速度快于基于LU分解的标准求逆方法，厄米特正定矩阵的Frobenius求逆速度快于基于Cholesky分解的标准求逆方法。我们开展了大量数值实验，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求逆的效率高于LU/Cholesky分解，且精度损失可忽略不计。一项附带成果是将高斯乘法推广到迭代二次扩域，我们证明该推广与用于快速整数乘法的Karatsuba算法以及多维快速傅里叶变换密切相关。
英文摘要：
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.
Precision-induced Adaptive Randomized Low-Rank Approximation for SVD and Matrix Inversion
Nanjing University of Information Science and Technology（南京信息工程大学）；Xi’an Jiaotong University（西安交通大学）；Xiamen University（厦门大学）；Pengcheng Laboratory（鹏城实验室）
Combinatorial and Recurrent Approaches for Efficient Matrix Inversion: Sub-cubic algorithms leveraging Fast Matrix products
Khalifa University of Science and Technology（Khalifa大学科学与技术）；Emirates Nuclear Technology Center (ENTC), Khalifa University of Science and Technology (KU)（阿联酋核技术中心（ENTC），Khalifa大学）
Theoretical Analysis of Thermodynamic Matrix Inversion: First-order Equivalence to Preconditioned Gradient Descent and Implications for Analog Computing
A fast methodology for large-scale focusing inversion of gravity and magnetic data using the structured model matrix and the $2D$ fast Fourier transform
Accelerated Stochastic Matrix Inversion: General Theory and Speeding up BFGS Rules for Faster Second-Order Optimization
Télécom ParisTech（巴黎电信学院）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；University of Edinburgh（爱丁堡大学）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；École polytechnique fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
Scalable parallel 3-D TEM inversion via rational approximation of the matrix exponential
Institute of Geophysics and Geoinformatics TU Bergakademie Freiberg（弗莱贝格工业大学地球物理与地信息学研究所）；Department of Mathematics The University of Manchester（曼彻斯特大学数学系）
Fast Matrix Inversion and Determinant Computation for Polarimetric Synthetic Aperture Radar
Graduate Program of Electrical and Computer Engineering Department, University of Calgary（卡尔加里大学电气与计算机工程研究生项目）；Signal Processing Group, Departamento de Estatística, Universidade Federal de Pernambuco（伯南布哥联邦大学统计系信号处理组）；Department of Electrical and Computer Engineering, University of Calgary（卡尔加里大学电气与计算机工程系）；Laboratório de Computação Científica e Análise Numérica, Universidade Federal de Alagoas（阿拉戈阿斯联邦大学科学计算与数值分析实验室）
On the Inversion of Polynomials of Discrete Laplace Matrices
University of Hasselt（哈瑟尔特大学）；Lancaster University（兰卡斯特大学）；Leiden University（莱顿大学）；Delft University of Technology（代尔夫特理工大学）；University of Johannesburg（约翰内斯堡大学）
