Matrices that are defective will not have a complete set of eigenvectors.
" In particular, an n × n matrix is defective if and only if it does not have n linearly independent eigenvectors."
The classic example that I know of is
>> A = triu(ones(2));
>> [V,D] = eig(A);
As you can see, there is a duplicate eigenvalue. But you can also see the two eigenvectors (columns of V) are not orthogonal.
In this case, the output from eig still satisfies the relation that A*V == V*D, at least within floating point trash.
>> norm(A*V - V*D)
How about the case of your matrix B?
>> [V,D] = eig(B);
>> norm(B*V - V*D)
So again, at best we can see this norm is zero. But V is not a complete set of eigenvectors. Why not? Because B is defective.
What can I say? If you read the help for eig, all it can promise is
"[V,D] = eig(A) produces a diagonal matrix D of eigenvalues and
a full matrix V whose columns are the corresponding eigenvectors
so that A*V = V*D."
When the matrix is defective, it can do no better than that, since the set of eigenvectors you are asking it to produce apparently don't exist.