repeated eigenvalue general solution problem

Hello all,

I am having trouble solving this repeated eigenvalue problem.

solving for the eigenvalues i get

which means

solving for the non-repeated solution using the eigenvalue

which gives me the eigenvector and solution

however when I try to solve for the repeated value

i get the matrix

I dont know how to find a LI eigenvector from this matrix. I have the solution from the back of my textbook, but even after just arbitrarily taking the same eigenvector that they chose, I cannot come up with the same solution. I believe the repeated solution should be of the form

where u and w are vectors, and u is actually the eigenvector from the matrix acquired using

I am stuck here, any help would be greatly appreciated.

Thanks,

Nick