I tried to solve a problem:how to find the jordan normal from of the following matrix.
Assume $n\ge 2$ , let $A=(a_{ij})_{n\times n}.$ $$A=\left( \begin{matrix} 0& 0& 0& 1& 1\\ 0& 0& \unicode{x22f0}& \unicode{x22f0}& 0\\ 0& 1& 1& 0& 0\\ 1& 1& 0& 0& 0\\ 1& 0& 0& 0& 0\\ \end{matrix} \right) \,\,,a_{ij}=\begin{cases} 1,i+j\in \left\{ n,n+1 \right\}\\ 0,i+j\notin \left\{ n,n+1 \right\}\\ \end{cases}.$$
I know $A$ is a symmetric matrix, so the jordan normal from of $A$ is a diagonal matrix. But computing the eigenvalues of $A$ may be difficult for me.
For example:
If n=3, the characteristic polynomial of $A$ is $\lambda^3-\lambda^2-2\lambda+1.$
If n=4, the characteristic polynomial of $A$ is $\lambda^4-\lambda^3-3\lambda^2+2\lambda+1.$
If n=5, the characteristic polynomial of $A$ is $\lambda^5-\lambda^4-4\lambda^3+3\lambda^2+3\lambda-1.$
I can't find a general conclusions about eigenvalues or even solve the above polynomials.
So I would like to know if there is a general conclusion about the eigenvalues of the above matrix or how to give the jordan normal form of $A$.
Thanks!