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 $A$?

Not much can be said. However, if $A$ is square and $X$ is Hermitian (hence $A$ is Hermitian and $C=B^\ast$) and $\lambda_1(M)\le\lambda_2(M)\le\lambda_3(M)\cdots$ denote the eigenvalues of a Hermitian matrix $M$ arranged in increasing order, we have the following interlacing inequality:
$$
\lambda_k(X)\le\lambda_k(A)\le\lambda_{k+n-r}(X)
$$
for $1\le k\le r$, when $A$ is $r\times r$ and $X$ is $n\times n$.
Also, if the four blocks have equal sizes, the characteristic polynomial of $X$ can be simplified as follows:
$$
\det\pmatrix{xI-A&-B\\ -C&xI} = \det(x^2I - xA - BC).
$$
Such simplification is valid because the two blocks at the bottom of the LHS commute. You can see that $\det(x^2I - xA - BC)$ has little resemblance to $\det(\lambda I-A)$ and we don't expect any relationship between the eigenvalues of $X$ and $A$ in general.
