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I have found the following characteristic polynomial: $$(x+2)(x-2)^2$$

I need to write down all the possible minimal polynomial, so I wrote: $${(x+2),(x-2),(x+2)(x-2),(x+2)(x-2)^2,(x-2)^2}$$

Why is it wrong?

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Firstly the minimal polynomial divides the characteristic polynomial which is what you seem to know already.

Secondly, any root of the characteristic polynomial must also be a root of the minimal polynomial so your first two options and the last one can't happen.

In short, you're allowed to reduce powers of the factors of the characteristic polynomial but the power still must be positive.

So the options are $(x+2)(x-2)$ and $(x+2)(x-2)^2$.

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  • $\begingroup$ What does it mean "but the still must be positive."? $\endgroup$ – gbox Aug 30 '15 at 10:46
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    $\begingroup$ I meant the power must be positive. If the characteristic polynomial is $(x-r_1)^{m_1} \cdots (x-r_n)^{m_n}$, with $r_i$ distinct, then the minimal polynomial can be anything of the form $(x-r_1)^{k_1} \cdots (x-r_n)^{k_n}$ with $1 \leq k_i \leq m_i$ for $i=1 \cdots n$. $\endgroup$ – Matt B Aug 30 '15 at 11:09

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