# Vectorization: Low Rank Matrix Factorization

- in the last video, we talked about Collaboration Filtering Algorithm
- This video, we talked about how's to do vectorization about Collaboration Filtering Algorithm
- Moreover,extending its benefit, which is find the similarity about other product that user purchased, and recommend it

- We already found table example above. Its telling us about the user's rating about each of the movie.
- What is this about, it's the same matrix Y. where y(i,j) is how user j rating about the movie i
- Remember, that the row of matrix Y is the movie, and the column is rating of the users.

- Now, we are going to vectorize the prediction implementation about matrix Y. which is y(i,j) is the matrix of user j rating about movie i
- If we see, the predicted ratings,we want to stack all the prediction computation which is done by matrix Y
- What we can do, first observe the matrix element of x and big thetas
- First we know that xi is a feature vector, and we want to make xi vector as one row. So we transpose the matrix xi
- theta also a vector. which is the rating of users based on all movies. Because we want thetai as one row, we also transpose thetas
- For both of all the element of matrix x and theta, we transpose each element, then we want to stack it into big matrix, X and big Theta
- Next to satisfy the calculation about user's prediction ratings, we use X mult bigTheta(t)
- Among the Linear Algebra Expert, this is what called Low Rank Matrix Factorization

- Now that we know all the feature in each of the product, we may want to know relatedness between them
- It's often difficult as it's more like human understanding about each product, but it certainly plaussible
- What might be the case in movie examples, is suppose the users purchase some products, and we may want to search of other similar products to recommend to them
- This is reasonable as we want users to keep engaging and purchasing our product
- So what we do is, we are trying to find the smallest distance about other products that user purchased, the smaller the distance is, then the two products is more related
- With this, we can find 5 of the smallest distance, and recommend 5 of most similar movies that we can recommend to the users

- By now, we already know about the vectorized implementation of Collaboration Filtering algoritm
- And we also learn how to use features of the movies, and find similar movies to recommend to users.