# Covariance vs Bandwidth of Kernel Density Estimate

I've been working with the scipy gaussian kde implementation (here), but I don't quite understand the difference between the bandwidth factor and the covariance matrix. I'm using it for a single dimension here, so the covariance matrix only contains a single value. The bandwidth factor is estimated using either Scott's or Silverman's method, and then apparently it is multiplied with the covariance matrix; but why? What does this covariance matrix actually represent? I understand the bandwidth is the 'width' of the kernel being used, but I don't see the role of the covariance matrix.

Scott's rule gives a bandwidth factor of $n^{- \frac{1}{d + 4}}$, where $n$ is the number of data points and $d$ their dimension; Silverman's is $(\frac14 n (d + 2))^{-\frac{1}{d + 4}}$.