This looks like standard case of floating point error. If we subtract the two results and use Chop we get:

{0, 0, 0, -9.9996*10^-7, 1.*10^-6, 6.60625*10^-10 + 5.08267*10^-9 I}

This is pretty close to a vector of 0s, so the results aren't far apart. I wouldn't say it's completely wrong. Do you believe that the floating point error is too high? If so are you mixing symbolic and numerical computations? Symbolic manipulations on floating point numbers aren't always the most numerical stable way to calculate something.

You can always improve the accuracy/precision of the results by staying with symbolic evaluation or using higher precision numbers.

Ok thanks, actually I need a better precision especially for the very small eigenvalues, do you know if quadruple precision is supported by mathematica ?

Hi thanks, actually there is a direct way to define with precision the parameters with `` as given in http://reference.wolfram.com/language/ref/Accuracy.html. If you also define a parameter called zero for 0 and the others like

a = 1``50;
b = 100``50;
c = 10^10``50;
zero = 0``50;

you get the result I expected, and the execution speed is not so bad.

Hi Jonathan, in fact the matrix is rather simple, so you can solve it as a symbolic matrix:

(mat = {{0, 0, 0, a, -I*a, b}, {0, 0, 0, a, -I*a, b}, {0, 0, 0, 
     a, -I*a, b}, {a, a, a, c, 0, 0}, {-I*a, -I*a, -I*a, 0, c, 0}, {b,
      b, b, 0, 0, c}}) // MatrixForm

{ev, evec} = Eigensystem[mat];
ev
{0, 0, c, c, 1/2 (c - Sqrt[12 b^2 + c^2]), 
 1/2 (c + Sqrt[12 b^2 + c^2])}

Replacing the symbols with exact numbers results in:

s1 = ev /. { a -> 1, b -> 100, c -> 10^10}
{0, 0, 10000000000, 10000000000, 
 1/2 (10000000000 - 200 Sqrt[2500000000000003]), 
 1/2 (10000000000 + 200 Sqrt[2500000000000003])}

Replacing the symbols with numeric arguments gives:

s2 = ev /. { a -> 1., b -> 100., c -> 1. 10^10}
{0, 0, 1.*10^10, 1.*10^10, -2.86102*10^-6, 1.*10^10}