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Give an example of two 5x5 matrices that have the same characteristic and minimal polynomials but they are not similar.

I've been thinking about this for a while but I can't find any example that satisfies the request. I would be grateful if someone could give me an example.

Thank you so much.

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Let's work over $\mathbb{C}$. Consider the Jordan canonical form

Hint: What does it mean that the minimal polynomials are the same?

For each eigenvalue $\lambda$, the maximal block has the same size.

Hint: What does it mean that the characteristic polynomials are the same?

For each eigenvalue $\lambda$, the dimensions of the generalized eigenspace are the same.

So, how do we find such an example?

Hint: $1+1+1+2 = 1 + 2 + 2$.

Consider Jordan blocks corresponding to $x \times x \times x \times x^2$ and $ x \times x^2 \times x^2$


Followup: Is 5 the smallest dimension for such an example?
Can you find an example in dimension 4?
Can you find an example in dimension 3?

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  • $\begingroup$ Your second hint is not correct as it is. We need the generalized eigenspaces to have equal dimensions. $\endgroup$ – Berci Dec 4 '19 at 9:12
  • $\begingroup$ Yes indeed. I've added that in. I thought that was obvious by talking about Jordan form. $\endgroup$ – Calvin Lin Dec 4 '19 at 15:40

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