Chapter 7

1. The covariance matrix is already diagonal; therefore, the eigenvectors are the standard x and y versors (1,0) and (0, 1), and the eigenvalues are 2 and 1. Hence, the x axis is the principal component, and the y axis is the second one.
2. As the ball B_0.5(0, 0) is empty, there are no samples around the point (0, 0). Considering the horizontal variance σ_x² = 2, we can imagine that X is broken into two blobs, so it's possible to imagine that the line x = 0 is a horizontal discriminator. However, this is only a hypothesis, and it needs to be verified with actual data.

3. No, they are not. The covariance matrix after PCA is uncorrelated, but the statistical independence is not guaranteed.
4. Yes; a distribution with Kurt(X) is super-Gaussian, so it's peaked and with heavy tails. This guarantees finding independent components.
5. As X contains a negative element,...