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A question in Dummit & Foote is asking to prove that two $3\times 3$ matrices are similar iff they have the same characteristic and the same minimal polynomial. I was able to prove that. But then the question is asking me to give an explicit counterexample to this assertion for $4\times 4$ matrices. And I do not know how to give this example.

Could anyone help me please?

Hope
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    Consider the case where the matrix has an eigenvalue of multiplicity 4. – user1551 Jan 14 '24 at 05:52
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    Read the 3rd comment here https://math.stackexchange.com/questions/4687946/show-that-two-matrices-are-similar-in-m-p-mathbbf-p – JimmyK4542 Jan 14 '24 at 05:54
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    From the duplicate: the matrices $$\left(\begin{array}{cccc} 0 & 1 & 0 & 0\ 0 & 0 & 0 & 0\ 0 & 0 & 0 & 1\ 0 & 0 & 0 & 0 \end{array}\right)\qquad\text{and}\qquad \left(\begin{array}{cccc} 0 & 1 & 0 & 0\ 0 & 0 & 0 & 0\ 0 & 0 & 0 & 0\ 0 & 0 & 0 & 0 \end{array}\right)$$ both have characteristic polynomial $x^4$ and minimal polynomial $x^2$, but since their Jordan canonical forms are different, they are not similar. – Dietrich Burde Jan 14 '24 at 17:19

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For example,choose $A=\left( \begin{matrix} 0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ \end{matrix} \right) $, $B= \left( \begin{matrix} 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \end{matrix} \right) $. The characteristic and minimal polynomial is $\lambda^4$ and $\lambda^2$.

But $A$ and $B$ are not similar.

fusheng
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