Imaginary axis eigenvalues of matrix delay equations with a certain alternating coefficient structure

In this paper we examine the imaginary axis eigenvalues of matrix delay differential equations with coefficients alternating between Hermitian and skew Hermitian, such as occurring in many mechanical systems. We show that given this special structure and a certain sign condition, the dimension of the eigenvalue problem is greatly reduced. We also give some theorems on eigenvalue crossing directions in this context, and show how reduction of dimension and crossing theorems work to provide insight on mechanical systems with time delay.