Man denkt an das, was man verließ; was man gewohnt war, bleibt ein Paradies (Johann Wolfgang von Goethe, Faust II, 1749–1832). We think of what we left behind; what we are familiar with remains a paradise.
Gaussian random vectors are special: uncorrelated Gaussian vectors are independent. The difference between independence and uncorrelatedness is subtle and is related to the deviation of the distribution of the random vectors from the Gaussian distribution.
In Principal Component Analysis and Factor Analysis, the variability in the data drives the search for low-dimensional projections. In the next three chapters the search for direction vectors focuses on independence and deviations from Gaussianity ...