  We consider a new class of non-self-adjoint matrices that arise from an
indefinite self-adjoint linear pencil of matrices, and obtain the spectral
asymptotics of the spectra as the size of the matrices diverges to infinity. We
prove that the spectrum is qualitatively different when a certain parameter $c$
equals $0$, and when it is non-zero, and that certain features of the spectrum
depend on Diophantine properties of $c$.
