Hi, I'm trying to think of appropriate tests for some software I developed for a student project that performs a PCA on some timeseries data. What I'd like to do is be able to "predict" in some way the form of the eigenvalue output by engineering the input time series properties. For example if I use Excel to create some uncorrelated return series (e.g. using NORMSINV(RAND()) with a resulting diagonal VCV, should I expect to get eigenvalues that are somehow related in magnitued to the original VCV? (Would the be the same?).What if I apply some constant decay factor to the VCV diagonal elements so that second element is some fraction of the first while the third is the same constant fraction of the second etc. - would I expect to see this pattern translated into the eigenvalues? Apologies in advance for my lack of knowledge of eigenvalues and matrix algebra.

Your question doesn't mean much if you're talking about uncorrelated series, as dimension reduction is not possible.

Its not true that you cannot perform dimensionality reduction of uncorrelated series. PCA will do exactly that when presented with multivariate Gaussian data. Whether or not it is meaningful is another matter. But it will certainly tell you about the variance of the data in the relevant dimensions and this can be useful regardless of correlation structure. In fact covariance will tend to decrease the efficacy of PCA in which case one must resort to such methods as kernel PCA.PCA is the eigensystem of the covariance matrixThe eigenvalues thus determined are equivalent to the diagonal entries of the diagonalised covariance matrixHence, for a diagonal matrix, the eigenvalues adopt the values of the diagonal elements.Intuitively, if you have non-uniform diagonal elements (i.e. variances) your data defines a N dimensional sausage.PCA simply aligns N orthogonal axes with that sausage so as to maximise the overlap of sausage and axes.