# Fast decay of eigenvector elements

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).

Can we find the matrix with the most compact eigenvector, without diagonalizing all of them? What I mean is that the eigenvector elements, if sorted in decreasing order based on their absolute values, decay fastest. An ideal situation would be to have the eigenvector [1, 0, ..., 0].