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Say I have a set of genomic data to be analysed for population structure. I ran two different analyses using two different maximum principal components. For the first analysis, I modeled the data using PCA up to the 10th dimension (i.e. up to PC10), and let's call this analysis A. The second analysis, still using the same data set, I modelled using PCA up to the 50th dimension (i.e. up to PC50). Let's call this analysis B. So now both analysis A and B respectively have 10 and 50 eigenvalues.

My question is: Do the 10 eigenvalues of analysis A equal to the first 10 eigenvalues of analysis B?

To put this into context, I'll describe a problem I encountered. When plotting PCAs for population genetics, up to 25 principal components are included in a study. The eigenvalues are as expected, where the last few eigenvalues explained minimal variance in the data. So, I'm wondering if it's worth the effort for me to do a separate PCA analysis to include higher dimensions (e.g. up to PC50). If the eigenvalues in the first 25 principal components remained the same, then running more dimensions will bring no relevant weight to the overall result.

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The answer is: yes, 10 eigenvalues of analysis A equal to the first 10 eigenvalues of analysis B.

Why is that? Eigenvalues are derived from covariance (or correlation) matrix of your data. This matrix is the same in both A and B analyses (since you have the same data in them). So the eigenvalues are the same too.

The only difference between analysis A and B is that in A you decide to look only at 10 largest (in absolute value) eigenvalues and "forget" about the rest while in B you look at 50 largest.

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