The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems, dynamic low-rank decompositions and model reduction. With this mostly expository work, we aim to provide a collection of the essential facts and formulae on the geometry of the Grassmann manifold in a fashion that is fit for tackling the aforementioned problems with matrix-based algorithms. Moreover, we expose the Grassmann geometry both from the approach of representing subspaces with orthogonal projectors and when viewed as a quotient space of the orthogonal group, where subspaces are identified as equivalence classes of (orthogonal) bases. This bridges the associated research tracks and allows for an easy transition between these two approaches. Original contributions include a modified algorithm for computing the Riemannian logarithm map on the Grassmannian that is advantageous numerically but also allows for a more elementary, yet more complete description of the cut locus and the conjugate points. We also derive a formula for parallel transport along geodesics in the orthogonal projector perspective, formulae for the derivative of the exponential map, as well as a formula for Jacobi fields vanishing at one point.

翻译：线性子空间的格拉斯曼流形在众多应用的数学建模中至关重要，涵盖机器学习、计算机视觉与图像处理中的问题，以及低秩矩阵优化、动态低秩分解和模型降阶等。本文主要以综述形式，系统整理格拉斯曼流形几何的基础事实与公式，旨在为上述问题提供基于矩阵算法的解决方案。此外，我们从两个视角阐述格拉斯曼几何：一是用正交投影算子表示子空间，二是将其视为正交群的商空间（子空间被识别为正交基的等价类）。这桥接了相关研究脉络，并便于两种方法间的灵活转换。原创贡献包括：提出一种改进的格拉斯曼流形上黎曼对数映射计算算法，该算法在数值上更具优势，同时能更基础、更完整地描述割迹与共轭点；从正交投影算子视角推导出沿测地线的平行移动公式、指数映射导数公式，以及在某点消逝的雅可比场公式。