Let A be a set of similar (symmetric) matrices, sharing the same eigenvalues. I understand that their eigenvectors would be different. Let us focus on one eigenvector (e.g. corresponding to the lowest eigenvalue).

How can we find the basis on which the eigenvector is most compact? What I mean is that the eigenvector elements, if sorted in decreasing order based on their absolute values, decay fastest.

Thanks in advance!

Edit: replaced PSD with symmetric to make the question more general.