Eigenvalues of a block matrix For $X=\left(\begin{array}{cc} A & B\\ C & 0\end{array}\right)$, how are eigenvalues of $X$ related to the eigenvalues of $A$?

Say that for some v, $ Av = \lambda v $ and $ C v = 0 $. Then letting $ w = \left(\begin{array}{c}
v \\
0 
\end{array}\right) $, what is $ X w $? Similarly if $ w = \left(\begin{array}{c}
x \\
y 
\end{array}\right) $ is an eigenvalue of $ X $ and $ B y = 0 $, then what is $ A x$?
