Fourier-Galerkin matrix for a drift-diffusion equation

For $u_t=u_{xx}-w^{\prime }{u}_x$ and Fourier truncation $|n|\le D$,whereFor $w(x)=\cos \pi x$, the nonconstant block has Gershgorin discs in the closed left half-plane, while the constant mode lies in the kernel. Hence every eigenvalue has nonpositive real part and the matrix is singular.