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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?

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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).

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  • $\begingroup$ Thanks for responding. I tried using the following program: "mm[x_, y_, z_] := With[{vals, vecs} = Eigensystem[M[x, y, z]], Pick[vecs, Positive[Re[vals]]]]; mm[0, 0, 0] // MatrixForm" where m[x,y,z] is 16 cross 16 matrix. But it shows an error : "With::lvlist: Local variable specification {vals,vecs}=Eigensystem[M[0,0,0]] is not a List." $\endgroup$ – atanu Sep 26 '18 at 9:06
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    $\begingroup$ You used With[] wrong. In this case, Module[] is more appropriate: mm[x_, y_, z_] := Module[{vals, vecs}, {vals, vecs} = Eigensystem[M[x, y, z]]; Pick[vecs, Positive[Re[vals]]]] $\endgroup$ – J. M.'s technical difficulties Sep 26 '18 at 9:16
  • $\begingroup$ Thanks a lot. Now it is working. $\endgroup$ – atanu Sep 26 '18 at 9:57

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