Suppose the characteristic polynomial is $x^4$. Is it possible to get a jordan block of size$ J(2)J(2)$?

Question

Suppose the characteristic polynomial is $x^4$. Is it possible to get a jordan block of size $ J(2)J(2)$?

This would mean that I get $2$ vectors in the eigenspaces, but then it seems like an impossible task

Answer 1

Do you mean $$ \pmatrix{0 & 1 & 0 & 0\cr 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 1\cr 0 & 0 & 0 & 0\cr} \ ?$$ What seems impossible?

Answer 2

The possibilities are $$J(1),J(1),J(1),J(1)$$ if the minimal polynomial is $x$, or $$J(2),J(1),J(1) \quad \textrm{ or} \quad J(2),J(2)$$ if the minimal polynomial is $x^2$, or $$J(3),J(1)$$ if the minimal polynomial is $x^3$, or $$J(4)$$ if the minimal polynomial is $x^4$.

Notice that all of these are possibilities since you could just write down any of these matrices and you would have a matrix whose Jordan form is what you were asking for.

For instance $$\begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$ is a matrix whose Jordan form (is itself) is of the form $J(3),J(1)$.

And the matrix in Robert's response has Jordan form of type $J(2),J(2)$.