%file is s12.tex
Now suppose that the hard case holds,
i.e. $B-\lambda^* C$ is singular. Equivalently, $D-\lambda^* S$ is
singular, where  $T^tBT = D$ and $T^tCT=S$ are both
diagonal, $T$ nonsingular.  Let $\bar{y} =
T(D-\lambda^*S)^{\dagger}T^{t}\psi$,
where $\dagger$ denotes the Moore-Penrose
generalized inverse. From the optimality conditions, we have that $\psi
\in R (B-\lambda^*C).$ Let $\lambda_k \rightarrow \lambda^*$ with
$B-\lambda_kC$ positive definite. Let
$\nu_k,\omega_k$ correspond to $\lambda_k$ as $\nu^*,\omega^*$
corresponds to $\lambda^*$. Then, from the simultaneous diagonalization,
we conclude that
\[
y_k =
(B-\lambda_kC)^{-1}\psi  = T(D-\lambda_kS)^{-1}T^{t} \psi \rightarrow
\bar{y}.
\]