See my following code snippet:

In[344]:= (*
https://mathematica.stackexchange.com/questions/16414/how-can-i-convert-a-complex-number-into-an-exponent-form
*)
polarForm = Expand[# /. z_?NumericQ :> Abs[z] Exp[I Arg[z]]] &;
polarForm1 = ComplexExpand[# /. z_?NumericQ :> Abs[z] Exp[I Arg[z]], TargetFunctions -> {Re, Im}] &;
x1=Eigenvectors[gen1]
polarForm/@#&/@x1
polarForm1/@#&/@x1

Out[346]= {{1, 1, 1}, {1/2 (-1 - I Sqrt[3]), 1/2 (-1 + I Sqrt[3]), 
  1}, {1/2 (-1 + I Sqrt[3]), 1/2 (-1 - I Sqrt[3]), 1}}

Out[347]= {{1, 1, 1}, {E^(-((2 I \[Pi])/3)), E^((2 I \[Pi])/3), 
  1}, {E^((2 I \[Pi])/3), E^(-((2 I \[Pi])/3)), 1}}

Out[348]= {{1, 1, 1}, {-(1/2) - (I Sqrt[3])/2, -(1/2) + (I Sqrt[3])/2,
   1}, {-(1/2) + (I Sqrt[3])/2, -(1/2) - (I Sqrt[3])/2, 1}}

Why doesn't the second method work?

Regards,
Zhao