The eigenvalues are the roots of the characteristic polynomial. Factored, that polynomial is (as a poly in x):
-(2 qd + 2 qv - x) x (4 qd qo + 4 qd qu + 16 qo qu + 4 qd qv +
4 qo qv + 4 qu qv - 2 qd x - 4 qo x - 4 qu x - 2 qv x +
x^2) (2 qd qo + 2 qd qu + 4 qo qu + 2 qd qv + 2 qo qv + 2 qu qv -
2 qd x - 2 qo x - 2 qu x - 2 qv x + x^2)^2
The problem at hand is to find relations amongst the matrix elements so that exactly four eigenvalues are zero. One is zero independent of the matrix element values. If you zero the quadratic of multiplicity 2 then the rank is 3, so the only option is to zero the remaining factors.
In[11]:= solns =
Solve[{{(2 qd + 2 qv - x), (4 qd qo + 4 qd qu + 16 qo qu + 4 qd qv +
4 qo qv + 4 qu qv - 2 qd x - 4 qo x - 4 qu x - 2 qv x +
x^2)} ==
0, (2 qd qo + 2 qd qu + 4 qo qu + 2 qd qv + 2 qo qv + 2 qu qv -
2 qd x - 2 qo x - 2 qu x - 2 qv x + x^2) != 0} /. x -> 0]
(* Out[11]= {{qu -> qd^2/(4 qo), qv -> -qd}} *)
