Another approach, which gives the additional eigenvector that you calculated, is to obtain the eigenvector(s), x, for eigenvalue v as solutions of (A-v*I)*x = 0:
format rational
A = [1 1 4 0; 1 1 1 -1; 0 0 3 1; 0 0 3 1];
display_eigenvectors(A)
function display_eigenvectors(A)
%DISPLAY_EIGENVECTORS Display eigenvectors for square matrix A.
% The eigenvector(s), x, for each eigenvalue, v, are obtained as
% a rational orthnormal basis of the null space of (A-v*I), where
% I is the unit matrix with the same size as A. The eigenvectors
% are then solutions of (A-v*I)*x = 0.
I = eye(size(A));
for v=unique(eig(A))'
fprintf(1,'\neigenvalue: %f\n eigenvector(s):\n',v)
disp(null(A-v*I,'rational'))
end
end
