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what is the Jordan normal form options from this minimal polynomial 𝑚𝐴(𝑥) = $𝑥^3 − 2x^2$. 4X4 matrix

I know of course that the 2 eigenvalues are 0,2.

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  • $\begingroup$ Can you tell what the algebraic multiplicities and geometric multiplicities of the two eigenvalues are? $\endgroup$ – Ted Shifrin Jul 8 '18 at 17:43
  • $\begingroup$ if i knew that i won't ask the question , need to find all the possible options. $\endgroup$ – Bentzi Rozen Jul 8 '18 at 17:44
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Hint:

The characteristic polynomial is a multiple of the minimal polynomial, and they have the same irreducible factors, we face only two possibilities: $$\chi_A(x)=x^3(x-2),\quad\chi_A(x)=x^2(x-2)^2.$$

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  • $\begingroup$ ok , i also get it , but what about the jordan form , we need geometric multiplicities , its right to say that for the first one with $x^3$ the possible geometric multiplicities is 3,1 2,1 or 1,1 ? $\endgroup$ – Bentzi Rozen Jul 8 '18 at 18:01
  • $\begingroup$ For the case with $x^3(x-2)$, there are are only two possible Jordan normal forms. Check which is a root of the minimal polynomial (this shouldn't be long with a product by blocks). $\endgroup$ – Bernard Jul 8 '18 at 18:24

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