It seems to have been missed that `NullSpace[]` is a built-in function:

    mat = {{9, -48, 4}, {36, 4, -33}, {32, 9, 36}}/49;

    evs = Eigenvalues[mat]
       {I, -I, 1}

    NullSpace[mat - # IdentityMatrix[3]] & /@ evs
       {{{-(18/13) + (14 I)/13, 12/13 + (21 I)/13, 1}},
        {{-(18/13) - (14 I)/13, 12/13 - (21 I)/13, 1}},
        {{1/2, -(1/3), 1}}}

Since `NullSpace[]` returns a list of vectors instead of a single vector (as it should), one might want to execute `Flatten[%, 1]` afterwards to get the eigenvectors as a list of vectors. Note that the result in this case matches the output of `Eigenvectors[mat]`.