The idea is to motivate the SVD for use in a recommender system.
Consider a matrix $A\in \mathbb{R}^{f\times u}$ where $A_{ij}$ caputures how user $j$ rates film $i$ (on a scale from 1-10, some entries may be missing).
Considering matrices $K=AA^T$, $L = A^TA$ what do $K_{ij}$ and $L_{ij}$ tell us?
What interpretations can you give for matrices $U$ and $V$ in the SVD $A=U\Sigma V^T$?
Riiight ... so
K = $AA^T$ = $U\Sigma^2U^T$, L = $A^TA = V\Sigma^2V^T$
With $K_{ij} = \langle A[i,:], A[j,:] \rangle$ being the dot product of all ratings for film $i$ with all ratings for film $j$.
Similarly, $L_{ij} = \langle A[:, i], A[:,j] \rangle$ being the dot product of all ratings from user $i$ with all ratings from user $j$.
That tells us ... what exactly? I expect it to amount to some kind of similarity measure, but beyond that, I have no idea.
As for $U$ and $V$ ... I know the first $r$ of them to be Eigenvectors of $K$ and $L$ for some $r$. I also know
$u_1,..,u_r$ is an orthonormal basis for the column space
$v_1,..v_r$ is an orthonormal basis for the row space
but that doesn't really tell me anything.
Yet have no idea if the notion of "row space" and "column space" even makes sense. What would the "span" of $A$'s columns / rows even mean?
I also have no idea what the nullspace of $A$ or $A^T$ would represent - if it represents anything at all - nor what
$u_{r+1},..,u_{m}$ being an orthonormal basis for $ker(A)$
or $v_{r+1},..,v_{n}$ being an orthonormal basis for $ker(A^T)$
implies - again, if anything at all.