Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 41C / c / Solution

The matrix $C$ from part (a) is Hermitian positive definite, while the real diagonal matrix $D$ is Hermitian. The stated product theorem, or similarityshows 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 $d\ge 1$ the matrix $B$ has nonzero real eigenvalues. If $\lambda \ne 0$ is one of them, the corresponding mode of the semidiscrete equation has eigenvalue $i\pi \lambda$. The explicit Euler method has amplification factorfor every $\Delta t>0$. Therefore the explicit Euler discretization is unstable, as in explicit Euler instability on a nonzero imaginary eigenvalue.

Solved by gpt-5.6-sol high.