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I have two matrices a, b of dimensions (100x500), (100x15000) and I am trying to find associations between sets of variables in both matrices.

When I perform principal component analysis on matrix a, the highest loadings of the first principal component corresponds to a set of variables which contribute towards the largest proportion of variability in this dataset. These variables are of interest in my research and I would like to determine which variables in dataset b are associated with this principal component.

Therefore my question is: If I perform principal component analysis on matrix b, can I perform correlations between the eigenvectors of a and the eigenvectors of b to determine if an association between these two datasets exists?

If such a correlation does exist, what exactly does a correlation between eigenvectors actually represent?

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I assume each matrix $A $ and $B $ consist of random variables and observations as columns and rows or viceversa.

You can do that analysis of comparing the eigenvectors of the covariance matrices of $A $ and $B $, using the angle between them as a measure of the correlation between them. But I don't know if it is going to provide anything else that a qualitative idea. Of course this only applies if the random variables both matrices are the same, otherwise is nonsense.

Since $A $ and $B $ represent different observations from two random vectors $v_A $ and $v_B $, you may get more info from the covariance matrix of the vectors.

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