Let us consider a block matrix of the form
$$A= \begin{bmatrix} -(k+\mu)I & B \\ kI & -(\gamma + \mu)I \end{bmatrix},$$
where $I$ is the $n\times n$ identity matrix, $\gamma, k$ and $\mu$ are positive constants and
$$B= \begin{bmatrix} a_1b_1& \ldots & a_1b_n \\ \vdots& \ddots& \vdots \\ a_nb_1& \ldots& a_nb_n \end{bmatrix}$$
with $a_i, b_i >0$ for all $i=1,\dots,n $.
Which are the eigenvalues of matrix $A$? Is there any way to evaluate them through the eigenvalues of the blocks of matrix $A$?
Thanks in advance.
Given vectors $\rm{a}, \rm{b} \in \mathbb R^n$, let
$$\rm{M} := \begin{bmatrix} -(\kappa + \mu) \rm{I}_n & \rm{a}\rm{b}^\top\\ \kappa \,\rm{I}_n & -(\gamma + \mu) \rm{I}_n \end{bmatrix}$$
whose characteristic polynomial is
$$\begin{aligned} \det \left( s \rm{I}_{2n} - \rm{M} \right) &= \det \begin{bmatrix} (s + \kappa + \mu) \rm{I}_n & - \rm{a}\rm{b}^\top\\ - \kappa \,\rm{I}_n & (s + \gamma + \mu) \rm{I}_n \end{bmatrix}\\ &= \det \left( (s + \kappa + \mu) (s + \gamma + \mu) \, \rm{I}_n - \kappa \, \rm{a}\rm{b}^\top \right)\end{aligned}$$
because multiples of the identity matrix do commute. Let $q$ be the characteristic polynomial of rank-$1$ matrix $\kappa \, \rm{a}\rm{b}^\top$. Hence,
$$q (s) = s^{n-1} \left( s - \kappa \, \rm{b}^\top \rm{a} \right)$$
and, thus,
$$\det \left( s \rm{I}_{2n} - \rm{M} \right) = q \left( (s + \kappa + \mu) (s + \gamma + \mu) \right)$$