We see that with just a vector of length 100, a scalar, and a vector of length 24, we actually come close to reconstructing the a 100 × 24 matrix. This is our first matrix factorization: We know it...

We see that with just a vector of length 100, a scalar, and a vector of length 24, we actually come close to reconstructing the a 100 × 24 matrix. This is our first matrix factorization:

We know it explains s$d[1]ˆ2/sum(s$dˆ2) * 100 percent of the total variability. Our approximation only explains the observation that good students tend to be good in all subjects. But another aspect of the original data that our approximation does not explain was the higher similarity we observed within subjects. We can see this by computing the difference between our approximation and original data and then computing the correlations. You can see this by running this code: