Some thoughts about matrix coordinate transformations

TL;DR

This paper explores matrix coordinate transformations as substitution operators, constructs tensor-like objects, discusses issues with contravariant vectors, and proposes using substitution operators to find inclusion functions.

Contribution

It introduces a novel perspective on matrix coordinate transformations as substitution operators and addresses problems in tensor generalizations.

Findings

Matrix transformations can be viewed as substitution operators.

Constructed objects that mimic tensor properties.

Identified issues with matrix generalization of contravariant vectors.

Abstract

Matrix coordinate transformations are defined as substitution operators without requiring an ordering prescription or an inclusion function from the Abelian coordinate transformations. We construct transforming objects mimicking most of the properties of tensors. We point out some problems with the matrix generalization of contravariant vectors. We suggest to use the substitution operators to search for an inclusion function.