matrices – Is there relation between eigenvalues of A,B,A+B,AB? Answer

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matrices – Is there relation between eigenvalues of A,B,A+B,AB?

I found there are some works about eigenvalues ​​of $A,B,A+B$ in the case of Hermitian matrices. For example, here.

I am wondering in the case that $A$ other $B$ is not Hermitian.

For example, my question explicitly regarding the following matrices:

$A=\begin{pmatrix} 0 & -1 & 0 & 0 & 1\\ 1 & 0 & -1 & 0 & 0\\ 0 & 1 & 0 & -1 & 0\\ 0 & 0 & 1 & 0 & -1\\ -1 & 0 & 0 & 1 & 0\\ \end{pmatrix}$

other

$B=\begin{pmatrix} 0 & 0 & 0 & 0 & 0\\ 2&-2&0&0&0\\ 0&2&-2&0&0\\ 0&0&2&-2&0\\ -2&0&0&2&0\\ \end{pmatrix}$

Note that $A$ is a circulating matrix, and $B$ is not rank-1 matrix.

Thanks for any suggestion!

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