  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.
