The system comes down to the equation x+y+z = 0, from which you can get two linearly independent vectors, e.g. (1,0,-1) and (1,-1,0).
I am having trouble solving this repeated eigenvalue problem.
solving for the eigenvalues i get
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.