  We prove that every element of the special linear group can be represented as
the product of at most six block unitriangular matrices, and that there exist
matrices for which six products are necessary, independent of indexing. We
present an analogous result for the general linear group. These results serve
as general statements regarding the representational power of alternating
linear updates. The factorizations and lower bounds of this work immediately
imply tight estimates on the expressive power of linear affine coupling blocks
in machine learning.
