The Change of Basis Groupoid

TL;DR

This paper explores the structure of change of basis matrices in finite vector spaces, revealing they form a connected groupoid, and introduces a method to algebraically express these matrices with applications to orthogonal polynomials.

Contribution

It demonstrates that change of basis matrices form a connected groupoid and proposes a novel algebraic expression method for these matrices, including practical examples.

Findings

Change of basis matrices form a connected groupoid of order m^2.

A general method to algebraically express change of basis matrices is developed.

Applications include examples with orthogonal polynomials.

Abstract

We show that the change of basis matrices of a set of $m$ bases of a finite vector space is a connected groupoid of order $m^2$. We define a general method to express the elements of change of basis matrices as algebraic expressions using optimizations of evaluations of vector dot products. Examples are given with orthogonal polynomials.