# Interpretation of a low-rank correlation matrix

I have a set of data from which I have generated a 12x12 Pearson correlation matrix. The rank of the correlation matrix is 8 but the number of eigenvalues that are of reasonable magnitude (not close to 0) is more like 3-4.

Is there some interpretation that can be made from the fact that the correlation matrix rank-and to a greater extent, the "effective" rank-is less than the size? I am struggling to determine if this tells me something additional about the underlying data or if it is just an accidental-and possibly common-occurrence.

• If you look at the correlation matrix does that show very high correlations? If so you have your answer. If not then we would need to understand your original variables to go any further I think. – mdewey Jul 10 '17 at 17:37
• @mdewey Yes, many of the correlations are quite high, although I don't see how that relates to my question. I want to know if there is more information I can glean aside from the usual interpretation of correlations. The data is forecasting different products at increasing lead times, with the correlation calculated between products' forecasts. – Cole Jul 10 '17 at 18:40
• Well, if all eigenvalues but four are very close to zero then the data points will be close to some 4D subspace. – kjetil b halvorsen Sep 15 '17 at 19:08