Question: Find a basis for each generalized eigenvalue of $\displaystyle L_A$ consisting of a union of disjoint cycles of generalized eigenvalues. Then find a Jordon canonical form $\displaystyle J$ of $\displaystyle A$. The matrix given is:

[11 -4 -5

21 -8 -11

3 -1 0]

My attempt: So the characteristic matrix is $\displaystyle f(x) = -(x+1)(x-2)^2$, which gives $\displaystyle x_1=-1,x_2=2$ with multiplicities of $\displaystyle 1,2$ respectively. Then $\displaystyle dim(K_1) = dim(E_1) = 1$ and $\displaystyle dim(K_2)=2$. Specifically, $\displaystyle K_1 = N(T+I)$ and $\displaystyle K_2 = N[(T-2I)^2]$. I managed to find a basis for $\displaystyle K_2$, which is

[1

3

0]

But when I tried it for $\displaystyle K_2$, I got:

$\displaystyle N[(T-2I)^2]$

[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!