How to collect eigenvectors corresponding to only positive eigenvalues?

Question

Let us consider a matrix of order $n \times n$ with $n/2$ positive and $n/2$ negative eigenvalues. How to collect $n/2$ eigenvectors corresponding to positive eigenvalues in a matrix of order $n \times n/2$? It can be done by arranging the eigenvalues in descending order and arranging the eigenvectors according to the new arrangement of corresponding eigenvalues. But how can one collect the eigenvectors corresponding to the positive eigenvalues without using descending order array of eigenvalues?

Answer 1

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).

Answer 2

I believe this should work (I have not extensively tested it),

system = Eigensystem[M]
systemSorted = Transpose@Sort@Transpose@system

The idea is to take the initial list of the form

system ~ { {e1, e2,...} , {{v11, v12,...}, {v21, v22,...}, ... } }

Transpose@system ~ { {e1, {v11, v12,...}}, {e2, {v21, v22,...}}, ... }

Sort then sorts these in increasing order of the eigenvalues, then Transpose takes you back to the original form of the list but now all eigenvalues and eigenvectors are sorted in ascending order.

Now we can simply take the last n/2 eigenvectors from the list. So overall

positiveEigensystem = (Transpose@Sort@Transpose@Eigensystem[M])[[-n/2;;]]

assuming n is even (e.g. I was working on a similar problem in which I knew for certain that the eigenvalues came in $\pm$ pairs).