Several recent papers have demonstrated the utility of using Lie groups within estimation problems, yielding improved accuracy and consistency. This paper introduces a new tool for describing operations with matrix Lie groups: tensors and the Einstein summation notation. While tensors and Einstein notation are well-known in other research fields, applying this mathematical notation to represent and compute matrix Lie derivatives is novel. More importantly, this new notation greatly clarifies the derivatives and operations necessary to work with matrix Lie Groups in (gradient-based) estimation frameworks. Therefore, the main contribution of this paper is not a new capability, but a more perspicuous mathematical notation for working with matrix Lie groups.

翻译：近期的多项研究证实了李群在估计问题中的有效性，显著提升了算法精度与一致性。本文提出一种描述矩阵李群运算的新工具：张量及爱因斯坦求和约定。尽管张量与爱因斯坦记号在其他研究领域已广为人知，但将其应用于矩阵李导数的表征与计算仍属创新。更关键的是，这种新记号能极大简化（基于梯度的）估计框架中矩阵李群相关导数与运算的表述。因此，本文的主要贡献并非提出新功能，而是为矩阵李群运算提供了更清晰直观的数学符号体系。