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

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.

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

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 $$1$$.