Determine the Jordan Normal Form over $\Bbb Q$ of the following matrix.
$$A=\begin{bmatrix}0&1&1&1&0\\0&2&0&0&0\\-1&3&0&0&0\\0&-1&1&0&1\\1&-1&0&-1&0\end{bmatrix}$$
The characteristic polynomial is $char_A(X)=(X-2)(X^2+1)^2$
Why is the number of Jordan block $0$ at $k=1, dim(ker(L_{A_2}^k))=2$???
Doesn't the number of blocks depends on the dimension of the kernel? i.e. because $dim(ker(A_2))=2$ there should be 2 Jordan blocks?