Question: Find a basis for each generalized eigenvalue of L_A consisting of a union of disjoint cycles of generalized eigenvalues. Then find a Jordon canonical form J of A. The matrix given is:
[11 -4 -5
21 -8 -11
3 -1 0]

My attempt: So the characteristic matrix is f(x) = -(x+1)(x-2)^2, which gives x_1=-1,x_2=2 with multiplicities of 1,2 respectively. Then dim(K_1) = dim(E_1) = 1 and dim(K_2)=2. Specifically, K_1 = N(T+I) and K_2 = N[(T-2I)^2]. I managed to find a basis for K_2, which is
[1
3
0]
But when I tried it for K_2, I got:
N[(T-2I)^2]
[9 -4 -5
21 -10 -11
3 -1 -2]^2 x = 0
which provides a basis with dimension 3, which makes no sense. Any hints, please? Thanks!