I have a square real matrix of the following block form:
$$M(k) = \left(\begin{array}{cc} 0 & A\\ kB & C \end{array}\right)$$
where $A$ is $1 \times m$, $B$ is $m \times 1$ and $C$ is $m \times m$, which makes $M$ an $n \times n$ matrix with $n = m+1$. Here $k$ is a scalar.
Is there a way to determine how the eigenvalues of $M$ depend on $k$? In other words, if I compute the eigenvalues of $M$ for $k = 1$, what can I say about the eigenvalues of $M(k)$ for other values of $k$?