Dear all,
after using the procedure gsl_eigen_hermv to diagonalize a matrix, I
suspect that the sign convention for the eigenvectors should be
reconsidered.
More specifically, I diagonalize a matrix which has only a few non-zero
entries, namely at m(2i,2i+1) and m(2i+1,2i) for all i. In my case, I
diagonalize a 8x8 matrix with real entries, e.g. 1.0. In other words,
the matrix represents the tensor product of the first Pauli matrix four
times with itself.
The eigenvectors should be of the following form (not normalized):
v1=(1,1,0,0,0....)
v2=(-1,1,0,0,0,...),
other eigenvectors can be obtained by shifting the non-zero coefficients
of v1 and v2 by two places to the right.
However, diagonalizing the 8x8 matrix described above, I obtain the
eigenvectors as described above exept the last one which is
v8=(0,0,0,0,0,0,1,-1) instead of (0,0,0,0,0,0,-1,1), i.e., multiplied
with (-1). Therefore, I think that the sign convention is either not
implemented correctly or the convention used is not broad enough.
Please find in the attachment to this e-mail a simple compilable code
which demonstrates this problem.
Thank you,
Walter
#include <iostream>
#include <gsl/gsl_math.h>
#include <gsl/gsl_eigen.h>
#include <gsl/gsl_complex.h>
#include <gsl/gsl_complex_math.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_blas.h>
int main()
{
int Ns = 3;
int Nwf = (int) pow(2,Ns);
//initialisation of the matrix
double Hamil[] = { 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0,
0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0,
};
gsl_matrix_complex_view Ham = gsl_matrix_complex_view_array(Hamil,Nwf,Nwf);
//print the matrix
{
std::cout << std::endl << "matrix" << std::endl;
for (int mp1=0;mp1<Nwf;mp1++)
{
for (int mp2=0;mp2<Nwf;mp2++)
{
printf("%5.2f",GSL_REAL(gsl_matrix_complex_get(&Ham.matrix,mp1,mp2)));
printf("%5.2fi ",GSL_IMAG(gsl_matrix_complex_get(&Ham.matrix,mp1,mp2)));
}
std::cout << std::endl;
}
}
gsl_vector *eval = gsl_vector_calloc (Nwf);
gsl_matrix_complex *evec = gsl_matrix_complex_calloc (Nwf,Nwf);
gsl_eigen_hermv_workspace * w = gsl_eigen_hermv_alloc (Nwf);
gsl_eigen_hermv (&Ham.matrix, eval, evec, w);
gsl_eigen_hermv_free (w);
//print the eigenvalues and eigenvectors of the matrix
{
for (int i = 0; i < Nwf; i++)
{
double eval_i = gsl_vector_get (eval, i);
gsl_vector_complex_view evec_i = gsl_matrix_complex_column (evec, i);
printf ("eigenvalue = %g\n", eval_i);
printf ("eigenvector = \n");
for (int j = 0; j < Nwf; ++j)
{
gsl_complex z = gsl_vector_complex_get(&evec_i.vector, j);
printf("%g + %gi\n", GSL_REAL(z), GSL_IMAG(z));
}
}
}
}
