Vtyjkyuk

Let $A$ be an $8times 8$ 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:

$J_1=beginpmatrix1&1&0&0&0&0&0&0\0&1&0&0&0&0&0&0\0&0&1&1&0&0&0&0\0&0&0&1&0&0&0&0\0&0&0&0&-2&1&0&0\0&0&0&0&0&-2&0&0\0&0&0&0&0&0&i&0\0&0&0&0&0&0&0&-iendpmatrix$, $J_2=beginpmatrix1&1&0&0&0&0&0&0\0&1&0&0&0&0&0&0\0&0&1&0&0&0&0&0\0&0&0&1&0&0&0&0\0&0&0&0&-2&1&0&0\0&0&0&0&0&-2&0&0\0&0&0&0&0&0&i&0\0&0&0&0&0&0&0&-iendpmatrix$

You identified the blocks and eigenspace dimensions correctly. The blocks for a given form, however, can be permuted. Thus, your first matrix represents 5!/2! = 60 forms, and your second matrix, 6!/2! = 360 forms for a total of 420 forms.