= Solution
The matrix $C$ from part (a) is <Hermitian positive-definite matrix>[Hermitian positive definite], while the real <diagonal matrix> $D$ is Hermitian. The stated product theorem, or similarity
$$
CD=C^{1/2}(C^{1/2}DC^{1/2})C^{-1/2},
$$
shows that every <eigenvalue> of $CD$ is real. Hence every eigenvalue of $B=-CD$ is real; this is the <real spectrum of a positive-Hermitian times Hermitian product>.
Moreover, $C$ is invertible and $D$ has rank $2d$, so for $d\geq1$ the matrix $B$ has nonzero real eigenvalues. If $\lambda\ne0$ is one of them, the corresponding mode of the semidiscrete equation has eigenvalue $i\pi\lambda$. The <explicit Euler method> has amplification factor
$$
G=1+i\pi\lambda\Delta t,
\qquad
|G|=\sqrt{1+\pi^2\lambda^2\Delta t^2}>1
$$
for every $\Delta t>0$. Therefore the explicit Euler discretization is unstable, as in <explicit Euler instability on a nonzero imaginary eigenvalue>.
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