  A generalization of the definition of an oscillatory matrix based on the
theory of cones is given in this paper. The positivity and simplicity of all
the eigenvalues of a generalized oscillatory matrix are proved. The classes of
generalized even and odd oscillatory matrices are introduced. Spectral
properties of the obtained matrices are studied. Criteria of generalized even
and odd oscillation are given. Examples of generalized even and odd oscillatory
matrices are presented.
