$Ch_A =(x+1)^6(x-2)^3 $ y $min_A = (x+1)^3(x-2)^2 $, List the possible Jordan forms for $A$

Question

Let $A$ be a complex matrix such that $Ch_A =(x+1)^6(x-2)^3 $ y $min_A = (x+1)^3(x-2)^2 $, List the possible Jordan forms for $A$. And in each case write the corresponding rational

I do not know if I'm wrong, but the possible Jordan forms are the intermediate polinomials.

$(x+1)^4(x-2)^2, (x+1)^5(x-2)^2, (x+1)^6(x-2)^2, (x+1)^4(x-2)^3,(x+1)^5(x-2)^3$

Answer 1

Jordan Forms are matrices, not polynomials.

Deciding on the JCF is the same as deciding on which Jordan blocks to use. And you already know a few facts about those:

1. You need to have Jordan blocks corresponding to the two eigenvalues of your matrix ($-1$ and $2$).

2. The sizes of your $-1$ Jordan blocks need to sum to $6$, and the sizes of the $2$ Jordan blocks need to sum to $3$ (because these are their multiplicities as roots of the characteristic polynomial).

3. The largest $-1$ block has size 3, and the largest $2$ block has size $2$ (because these are their multiplicities as roots of the minimal polynomial).

Any combination of blocks that satisfies these three conditions could potentially be a JCF for the matrix you describe.

So, for instance, you could have $2$-blocks of sizes $2$ and $1$, and $(-1)$-blocks of sizes $3$ and $2$. This would correspond to a JCF of $$ \left[\begin{array}{cc|c|ccc|cc} 2&1&0&0&0&0&0&0\\ 0&2&0&0&0&0&0&0\\\hline 0&0&2&0&0&0&0&0\\\hline 0&0&0&-1&1&0&0&0\\ 0&0&0&0&-1&1&0&0\\ 0&0&0&0&0&-1&0&0\\\hline 0&0&0&0&0&0&-1&1\\ 0&0&0&0&0&0&0&-1 \end{array}\right]. $$ (Bars added to emphasize the blocks.)

Can you see how to list all configurations in terms of the sizes of $2$-blocks and $(-1)$-blocks? As a hint, the only possible configurations for the $2$-blocks is to have a block of size $2$ and a block of size $1$.