A matrix method for determining the imaginary axis eigenvalues of a delay system

The author considers autonomous neutral or retarded matrix delay differential systems. The imaginary axis eigenvalues of such a system are shown to be contained in the set of generalized eigenvalues of an associated matrix pair. If the system is not a singular neutral system, one can replace generalized eigenvalues by the eigenvalues of a single matrix. We also show in both the neutral and retarded cases that system pure imaginary eigenvalues are limited to the values at which an associated second degree matrix polynomial becomes singular. Associated system eigenvectors are also eigenvectors of this matrix polynomial. Examples are given in which the matrix, the matrix pair, and the matrix polynomial are used for stability analysis.