[Help-gsl] Same eigenvector but different eigenvalues - Why?

[Antonio Divisato]

*Hi, I'm italian student. First of all sorry for my english.
I have a problem when I use GSL to test an example on classical scaling:
I have this matrix *
A =

   20.52000    1.64000  -18.08000   -4.09000
    1.64000   -0.83000    2.05000   -2.87000
  -18.08000    2.05000   11.39000    4.63000
   -4.09000   -2.87000    4.63000    2.33000

*and I want the eigendecomposition. So I write this code:*

int main(void) {

    int i=0, j=0;
    double nm;

gsl_vector *eval;
gsl_matrix *evec;

    gsl_matrix *matr=gsl_matrix_alloc(4,4);
    gsl_matrix_set(matr,0,0,20.52);
    gsl_matrix_set(matr,0,1,1.64);
    gsl_matrix_set(matr,0,2,-18.08);
    gsl_matrix_set(matr,0,3,-4.09);

    gsl_matrix_set(matr,1,0,1.64);
    gsl_matrix_set(matr,1,1,-0.83);
    gsl_matrix_set(matr,1,2,2.05);
    gsl_matrix_set(matr,1,3,-2.87);

    gsl_matrix_set(matr,2,0,-18.08);
    gsl_matrix_set(matr,2,1,2.05);
    gsl_matrix_set(matr,2,2,11.39);
    gsl_matrix_set(matr,2,3,4.63);

    gsl_matrix_set(matr,3,0,-4.09);
    gsl_matrix_set(matr,3,1,-2.87);
    gsl_matrix_set(matr,3,2,4.63);
    gsl_matrix_set(matr,3,3,2.33);


   eval = gsl_vector_alloc(4);
   evec = gsl_matrix_alloc(4,4);
    gsl_eigen_symmv_workspace *w = gsl_eigen_symmv_alloc(4);
    gsl_eigen_symmv(matr,eval,evec,w);
    gsl_eigen_symmv_free (w);

   gsl_eigen_symmv_sort(eval,evec,GSL_EIGEN_SORT_ABS_ASC);

for (i = 0; i < 4; i++)
           {
             double eval_i
                = gsl_vector_get (eval, i);
             gsl_vector_view evec_i
                = gsl_matrix_column (evec, i);

             printf ("eigenvalue = %g\n", eval_i);
             printf ("eigenvector = \n");
             gsl_vector_fprintf (stdout,
                                 &evec_i.vector, "%g");
           }
return 0;
}

The problem is that I have this result:
eigenvalue = -3.28845e-15
eigenvector =
-0.5
-0.5
-0.5
-0.5
eigenvalue = -6.53416
eigenvector =
0.0405866
-0.614856
-0.190051
0.76432
eigenvalue = 11.1461
eigenvector =
0.387874
-0.609699
0.586947
-0.365121
eigenvalue = -71.4219
eigenvector =
0.773244
0.0147966
-0.607761
-0.180279
*
While my book and OCTAVE says:*

octave-3.0.0:3> A=[20.52 1.64 -18.08 -4.09; 1.64 -0.83 2.05 -2.87; -18.08
2.05 11.39 4.63; -4.09 -2.87 4.63 2.33]
A =

   20.52000    1.64000  -18.08000   -4.09000
    1.64000   -0.83000    2.05000   -2.87000
  -18.08000    2.05000   11.39000    4.63000
   -4.09000   -2.87000    4.63000    2.33000

octave-3.0.0:4> [EVECT,EVAL]=eig(A)
EVECT =

  -0.388034   0.499844  -0.041102  -0.773237
   0.609619   0.500918   0.614189  -0.014765
  -0.587093   0.500013   0.189536   0.607770
   0.364851   0.499223  -0.764957   0.180282

EVAL =

   -5.56580    0.00000    0.00000    0.00000
    0.00000   -0.00750    0.00000    0.00000
    0.00000    0.00000    3.26739    0.00000
    0.00000    0.00000    0.00000   35.71592

*So I have the same eigenvector but different eigenvalues. Why? Can you help
me?*

Bye
Antonio Divisato