dimensionality of our space will be reduced from v to v — 1. If two variables are very highly correlated, then the elimination of one of them would produce only a slight loss of information, while the dimensionality of the space would be reduced to v — 1. So, one can deduce that the information contained in the "lost" vth dimension was well below the average of the information contained in the other dimensions.
It is quite apparent now that not all the dimensions have the same importance, and that, owing to the correlations among the variables, the "real" dimensionality of our data matrix is somehow lower
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