It seems like you have a representation of a graph using a weighted adjacency matrix.
If I understand correctly, you wish to have a single entry per edge.
You can do this by taking only the upper triangle of the adjacency matrix
A = sparse( A(:,1), A(:,2), A(:,3), max(A(:,2)), max(A(:,2)) );
[ii jj wij] = find( triu( A ) ) ;
disp( [ii jj wij] )
outputs:
1 2 0.563826
1 3 1.183216
2 3 0.830602
1 4 1.956860
2 4 1.651969
3 4 0.821523
2 5 1.771722
3 5 1.127165
3 6 1.781179