A note on unimodular eigenvalues for palindromic eigenvalue problems

We consider the occurrence of unimodular eigenvalues for palindromic eigenvalue problems associated with the matrix polynomial Pn(λ) ≡ ∑n i=0Aiλ i where Ai = An−i with M∗ ≡ M , M or PMP (P 2 = I). From the properties of palindromic eigenvalues and their characteristic polynomials, we show that eigenvalues are not generically excluded from the unit circle, thus occurring quite often, except for the complex transpose case when Pn is complex and M∗ ≡ M . This behaviour is observed in numerical simulations and has important implications on several applications such as the vibration of fast trains, surface acoustic wave filters, stability of time-delay systems and crack modelling.