How to collect eigenvectors corresponding to only positive eigenvalues?

The method proceeds in two stages: use Eigensystem[] to compute the set of eigenvalues and eigenvectors, and then use Pick[] to retain the eigenvectors corresponding to the positive eigenvalues.

As a concrete example, take the Clement-Kac matrices:

ckmat[n_Integer?Positive] :=
SparseArray[{{i_, j_} /; Abs[i - j] == 1 :>
             With[{k = Min[i, j]}, Sqrt[k (n - k)]]}, {n, n}]

Get the eigensystem for the $8\times 8$ case:

{vals, vecs} = Eigensystem[N[ckmat[8]]];

The desired eigenvector set can then be obtained as

Pick[vecs, Positive[vals]]

You can check that the right set was picked out by comparing the result of that with vals and vecs (keeping in mind that the eigenvectors are stored by rows).