This paper provides a comprehensive description of a new method of factorization for the Coriolis/centripetal matrix. In the past three decades, studies on dynamics have rapidly developed through the efforts of many researchers in the field of mechanics. While direct methods for deriving the Coriolis/centripetal matrix are well known and have been widely used in the last century, the entries of this matrix were always obtained by means of the Christoffel symbols of first kind. Startling techniques for deriving dynamic equations of robot manipulators first appeared about 30 years ago. Since then, much has been done to refine and develop the method, but it is still a highly active field of research, with many outstanding problems, both theoretical and in applications. This work presents, in a unitary frame and from a new perspective, the main concepts and results of one of the most fascinating aspects of mechanics, namely the factorization of structures, and offers the reader another point of view concerning a possible way to approach the Coriolis/centripetal matrix. It aims to study a theory of representation for such a matrix based on an elegant method of fundamental matrices. The paper is intended to be self-contained by presenting complete properties emerging from these novel structures. This work is useful not only to researchers in mechanics, but also to control engineers who are interested in learning some of the mechanical modeling. Toward this end, the paper provides numerical examples, as well as practical adaptive applications for modern designers to use at the system level.