A Cloud With No Favourite Direction

The dimensionality-reduction pipeline — centre the data, build the covariance matrix, find its eigenvectors, project onto them — hands you new coordinates that are no longer correlated with each other. But they still have very different spreads: the coordinate along PC1 varies a lot, the one along PC2 far less.

Whitening adds one last move. Divide each new coordinate by its own standard deviation, so that every direction ends up with variance exactly $1$. The cloud comes out round, with no direction more stretched than any other.

Task: write whiten(points) for 2D data. points is a list of $[x, y]$ pairs. Return the whitened pair for every point, in the same order, with each number rounded to 4 decimal places.

1. Centre. Subtract the mean point from every point, giving centred pairs $(d_x, d_y)$.
2. Covariance. With $n$ points, and dividing by $n$:

1. Eigenvectors. Find the eigenvalues of $C$, $\lambda_1 > \lambda_2$, and a unit eigenvector for each: $\mathbf{u}_1$ for $\lambda_1$ and $\mathbf{u}_2$ for $\lambda_2$. Flipping an eigenvector gives another eigenvector, so to make the answer unique, orient each one so that its first component is positive — or, if its first component is $0$, so that its second component is positive.
2. Rotate, then rescale. A centred point $\mathbf{d}$ becomes

You may assume there are at least three points, they don't all lie on one straight line, and the two eigenvalues are different — so both are positive, and each has a single eigenvector direction.

Example.

Here $C = \begin{bmatrix} 5 & 4 \\ 4 & 5 \end{bmatrix}$, with eigenvalues $9$ and $1$. Whatever the input, each output coordinate should come out with mean $0$ and variance $1$ — a handy check on your own work.