# Looking for two 5x5 matrices that have the same characteristic and minimal polynomials but they are not similar

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.

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?

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