  We study the optimization landscape of deep linear neural networks with the
square loss. It is known that, under weak assumptions, there are no spurious
local minima and no local maxima. However, the existence and diversity of
non-strict saddle points, which can play a role in first-order algorithms'
dynamics, have only been lightly studied. We go a step further with a full
analysis of the optimization landscape at order 2. We characterize, among all
critical points, which are global minimizers, strict saddle points, and
non-strict saddle points. We enumerate all the associated critical values. The
characterization is simple, involves conditions on the ranks of partial matrix
products, and sheds some light on global convergence or implicit regularization
that have been proved or observed when optimizing linear neural networks. In
passing, we provide an explicit parameterization of the set of all global
minimizers and exhibit large sets of strict and non-strict saddle points.
