How to update item and user factors ALS in Group Specific Recommendation?

I was going through this Group Specific Recommendation System paper. I want to implement this from scratch. I see that they have used Alternating Least Square. But how are they updating the item factors and user factors? Do I need to find the gradient of those equations (5), (6), (7), (8)?

The algorithm I am talking about is in 3.2.

Someone help me visualize this via an example of how is the calculation happening. Let me give a short example.

$Users \times Items$ matrix-

[[4, 5, NaN],
 [1, NaN, 2],
 [NaN, 1, 3]]

User latent factors ($U = Users \times K$). In the mentioned paper it is $P$. $K = latent\ factor\ dimension = 2$

[[0.5,  1],
 [-2,   3],
 [-1.5, 4]]

Item latent factors ($M^T = K \times Items$). In the mentioned paper it is $Q$. $K = latent\ factor\ dimension = 2$

[[0.5,  2, -3],
 [1.5,  3, -2]]

$S$

[[1,   -1],
 [-4, 0.4],
 [-5.5, 4]]

$T$

[[0.5,  1,    -1],
 [-1,   0.5, 2.5]]

How is the calculation with this dummy example?