# Recommendation matrix as a product of User Similarity and Ratings

For both item-item and user-user collaborative filtering the recommendation matrix $Γ_{m x n}$, which is an (m x n) matrix, can be defined as:

$$Γ(i,j)=r_{ij}$$ For user-user collaborative recommendation r_ij is defined as

$$r_{ij}= ∑_{(x∈ users)}[cosSim(x,i)*R_{xj} ]$$

For item-item collaborative recommendation $r_{ij}$ is defined as

$$r_{ij}= ∑_{(x ∈ items)}[R_{ix}*cosSim(x,j) ]$$

The matrix Γ can thus be defined as:

• for user-user filtering: $Γ= S_U * R$
• for user-user filtering $Γ= R * S_I$

In which scenarios does the equation Γ holds true. Assuming that we need to pick the k items with highest $r_{ij}$ in order to recommend the top-k items for a user i

• Try looking into SVD. It should work for what you want. – Anshul G. Feb 27 '17 at 12:29