Let $A$ be an $8\times 8$ complex matrix with characteristic polynomial $$p_A(x)=(x-1)^4(x+2)^2(x^2+1)$$ and minimal polynomial $$m_A(x)=(x-1)^2(x+2)^2(x^2+1).$$ Determine all possible Jordan canonical forms of $A$. Also, what is the dimension of the eigenspace for $\lambda = 1$ for each case?
I haven't learned minimal polynomial yet, so I wanted to check if I am on the right track.
From the characteristic polynomial, the eigenvalues are $i,-i,-2,1$. From the minimal polynomial, the block sizes are $1,1,2$ for eigenvalues $i,-i,-2$ respectively. For $\lambda = 1$, the largest block size is 2, so the possible block sizes are $2,2$, which has eigenspace dim of 2 ($J_1$). Or the sizes are $2,1,1$, which has eigenspace dim of 3 ($J_2$). Writing the full Jordan matrix:
$J_1=\begin{pmatrix}1&1&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&1&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&-2&1&0&0\\0&0&0&0&0&-2&0&0\\0&0&0&0&0&0&i&0\\0&0&0&0&0&0&0&-i\end{pmatrix}$, $J_2=\begin{pmatrix}1&1&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&-2&1&0&0\\0&0&0&0&0&-2&0&0\\0&0&0&0&0&0&i&0\\0&0&0&0&0&0&0&-i\end{pmatrix}$
