The Weighted Matrix Product is a generalization of the ordinary matrix product, which is useful in creating algebras with desired characteristics.

As a non-trivial illustration of its use, Quaternions are constructed from a simple set of 4 dimensional spanning matrices. The Cayley table defining the Quaternions is used to determine the weight matrices.

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Anyone familiar with electromagnetics and the theory of a complex variable must have wondered if there was some kind of "analyticity condition" in our normal space from which the field equations in our space arise. As it turns out, things are even simpler than that. Given a definition of "charge-current-density"
consistent with current definitions, a vector-geometry R-algebra may be constructed using the weighted matrix product where all smooth functions in the space satisfy Maxwell's equations; along with the kickers that the usual cross product arises naturally, and the dot product is the extension (component) in the time-dimension. Once the reader is comfortable with the weighted matrix product, she may find the mathematical examination of this R-algebra enlightening.

Through this ebook, the reader will see passed the illusions of separating geometry from physics, algebra from analysis; to the next thought leap discovering a more complete view of  'what is'.