This work aims to numerically construct exactly commuting matrices close to given almost commuting ones, which is equivalent to the joint approximate diagonalization problem. We first prove that almost commuting matrices generically have approximate common eigenvectors that are almost orthogonal to each other. Based on this key observation, we propose a fast and robust vector-wise joint diagonalization (VJD) algorithm, which constructs the orthogonal similarity transform by sequentially finding these approximate common eigenvectors. In doing so, we consider sub-optimization problems over the unit sphere, for which we present a Riemannian quasi-Newton method with rigorous convergence analysis. We also discuss the numerical stability of the proposed VJD algorithm. Numerical examples with applications in independent component analysis are provided to reveal the relation with Huaxin Lin's theorem and to demonstrate that our method compares favorably with the state-of-the-art Jacobi-type joint diagonalization algorithm.

翻译：本文旨在数值构造逼近给定几乎交换矩阵的精确交换矩阵，这等价于联合近似对角化问题。我们首先证明几乎交换矩阵通常具有近似正交的近似公共特征向量。基于这一关键观察，我们提出一种快速鲁棒的向量式联合对角化（VJD）算法，该算法通过依次寻找这些近似公共特征向量来构造正交相似变换。在此过程中，我们考虑单位球面上的子优化问题，并给出具有严格收敛性分析的黎曼拟牛顿方法。我们还讨论了所提出的VJD算法的数值稳定性。通过独立成分分析应用中的数值算例，揭示了与林华新定理的联系，并证明我们的方法相比最新型的雅可比型联合对角化算法具有更优性能。