The singular value decomposition (SVD) transforms the data matrix in a way that exposes the amount of variation in the data relative to a set of latent features. The most natural interpretation is geometric: given a set of data in m-dimensional space, transform it to a new geometric space in which as much variation as possible is expressed along a new axis, as much variation independent of that is expressed along an axis orthogonal to the first, and so on. In particular, if the data is not inherently m-dimensional, its actual dimensionality (the rank of the data matrix, A) is also exposed.