I started with the above matrix and found that the eigenvalues were 2 and 4.
When I was trying to calculate the base for for lambda=4, I came up with (0,1,0)^T and (0,0,1)^T. What bothers me is that the algebraic multiplicity in this case is 1, right? But the geometric multiplicity appears to be 2. This can't be because the algebraic multiplicity must be greater than or equal to the geometric multiplicity.
So where have I gone wrong in my reasoning?