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}