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