I don't know if this question is considered research-related. If not, I will move it to [Math SE](https://math.stackexchange.com). 




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I am searching for matrices with the property


$$|A|\_F^2 = \deg( \chi\_A(t) ) = 2 \deg( m\_A(t)), tr(A) = 1$$


where $\chi\_A(t)$ is the characteristic polynomial and $m\_A(t)$ is the minimal polynomial and $|.|\_F$ is the Frobenius norm. For some $n \in \mathbb{N}$, I have already found a matrix $A\_n$ with this property, but now I am asking myself how to construct or find other matrices with this property.


The matrix $A\_n$ for a natural number $n$ is defined as $A\_n = \oplus\_{d|n} Z\_d$ where $Z\_d$ is a circulant matrix with $0/1$ which is defined on Wikipedia as $Z$. I can show that


$$\chi\_A(t) = \prod\_{d|n}{t^d-1}=\prod\_{d|n} \phi\_d(t)^{f\_n(d)}$$


Hence one can conclude from this that:


$$\sigma(n) = \deg(\chi\_A(t)) = \sum\_{d|n} {f\_n(d) \phi(d)}$$


where $\phi$ is the Euler phi function. The minimal polynomial of $A\_n$ is $m\_A(t) = t^n-1$. For $f\_n(d)$ I make the conjecture that:
$$f\_n(d) = (a\_1-k\_1+1)\cdots(a\_r-k\_r+1)$$ 
where $n = p\_1^{a\_1}\cdots p\_r^{a\_r}$ is the prime decomposition of $n$ and $d= p\_1^{k\_1}\cdots p\_r^{k\_r}$ with $0 \le k\_i \le a\_i$ is a divisor of $n$.
It is clear that $|A\_n|\_F ^2 = \sigma(n)$
If $n$ is a perfect number we have


$$|A\_n|\_F^2 = \deg( \chi\_{A\_n}(t) ) = 2 \deg( m\_{A\_n}(t))$$ 


that is why I am searching for other matrices $A$ with this property and $tr(A)=1$ since we always have $tr(A\_n)=1$. Maybe this can shed some light on how difficult it is to have this property.
Also if somebody has an idea on how to prove the conjecture for $f\_n(d)$ that would also be fine.



Searching for matrices with some property
Take $J \oplus O$, where $J$ is the Jordan block with eigenvalue $0$ and size $n$, and $O$ is the $n\times n$ zero matrix. Replace $n+1$ zeros with ones anywhere in the upper triangular part of $J$ (which does not change the fact that $J^n=0$, $J^{n-1}\neq 0$). 


