The Representation of Matrices in Unit Vector Notation

Abstract

The objective of this paper is to express a matrix of any dimension in unit vector notation. This is accomplished by first solving the two and three dimensional cases before solving the general n-dimensional case. The fact that matrices can be represented as a (non-linear) combination of standard basis unit vectors shows that matrices are not simply abstract entities used just for representing data, but also have a geometric interpretation. This paper then defines the natural product and explains its applications in matrix calculus, before relating it to the outer product of real vectors, a special case of the tensor product. We then conclude that the space is the intersection of the exterior space and the newly defined \textit{natural space}.