  Several recent methods used to analyze asymptotic stability of
delay-differential equations (DDEs) involve determining the eigenvalues of a
matrix, a matrix pencil or a matrix polynomial constructed by Kronecker
products. Despite some similarities between the different types of these
so-called matrix pencil methods, the general ideas used as well as the proofs
differ considerably. Moreover, the available theory hardly reveals the
relations between the different methods.
  In this work, a different derivation of various matrix pencil methods is
presented using a unifying framework of a new type of eigenvalue problem: the
polynomial two-parameter eigenvalue problem, of which the quadratic
two-parameter eigenvalue problem is a special case. This framework makes it
possible to establish relations between various seemingly different methods and
provides further insight in the theory of matrix pencil methods.
  We also recognize a few new matrix pencil variants to determine DDE
stability.
  Finally, the recognition of the new types of eigenvalue problem opens a door
to efficient computation of DDE stability.
