Is it possible to have the details of the computation of the optimal expressions of Z^T and W^T, obtained by computing the gradient of the loss wrt to Z and W respectively ( from "Alternative Least-squares" part from lecture 13_a).
Thank you in advance.
You compute the gradient with respect to X and Z then set them to zero, after transposing you will get the expressions above.
All of this can be translated into the ALS algorithm which fixes one matrix and optimizes over the other one. I gave a detailed answer of how to get the update of X in an other question that was asked in this forum, I invite you to refer to it.