Let $A$ be a (symmetric) positive definite matrix and $\hat{n}$ be an arbitrary unit vector. Consider $b,c,d,k$ arbitrary positive integers. I would like to know if the following matrix has real eigenvalues (I would like the more general answer, for more than product of four matrices but even for three I don't know).
$$(I - \hat{n}\hat{n}^T)\cdot A^b\cdot(I - \hat{n}\hat{n}^T)\cdot A^c\cdot (I - \hat{n}\hat{n}^T)\cdot A^d\cdot (I-\hat{n}\hat{n}^T)\cdot A^k\cdot (I-\hat{n}\hat{n}^T).$$
I would like to note that I have asked the same question at mathstackexchange.
