During the early part of the 20^th Century, Max Planck proposed that the emission and absorption of light takes place in packets or quanta. The relationship advanced was that the energy of the emitted/absorbed radiation, was related to its frequency via a simple constant of proportionality, subsequently termed Planck's constant. Einstein then extended this idea as inherent in the very nature of all propagating electromagnetic radiation. Subsequently, Louis de Broglie, (pronounced "de Broy"), went further and postulated that this concept applied to all matter of whatever nature. The hypothesis advanced by de Broglie, in concert with Planck's relationship, was that all matter possessed a "matter wave", such that its inverse wavelength was related to mechanical momentum via Planck's constant. This paper investigates the characteristics of de Broglie's matter waves within Pseudo-Euclidean Space-Time, as represented by the Relativistic Space-Time Domain D₀, (See [1]). It will thereby provide a simple theoretical demonstration that matter, when possessing a spatial velocity within such a Domain, exists simultaneously in dual form - with both a particulate and a wave function existence. This then enables a number of concepts with regard to the nature of matter energy to be investigated. It will also provide the means by which the resurrection of the Bohr/Sommerfeld old quantum theory of atomic structure can be effected, and to provide that theory with an equal or better theoretical basis, to that of the modern quantum mechanics/quantum electrodynamics theory. This particular concept will form the subject of a future series of papers. This paper will include mathematical development in relativistic spatial/temporal form. Ref. [1] is particularly important in understanding such developments, and should therefore be read thoroughly first.