Let $p,a_1,...,a_n\in(0,1)$ and $\sum_{i=1}^na_i=1$. Now consider the following matrix:
$$ \left(\begin{array}{ccccc} (1-p) & \sqrt{p(1-p)}a_1 & \sqrt{p(1-p)}a_2 & ... & \sqrt{p(1-p)}a_n\\ \sqrt{p(1-p)}a_1 & pa_1 & 0 & ... & 0\\ \sqrt{p(1-p)}a_2 & 0 & pa_2 & ... & 0\\ ... & 0 & 0 & ... & 0\\ \sqrt{p(1-p)}a_n & 0 & 0 & ... & pa_n\\ \end{array}\right) $$
Some simple properties are that its rank is $n-1$ and for $a_i=1$ it has a single eigenvalue which equals 1. What I would like to know is a closed form for its eigenvalues. Does anybody see an easy way of computing the eigenvalues of this matrix?