# Generalisations of the determinant to interdimensional transformations: a review

## TL;DR

This paper reviews three generalisations of the determinant for transformations between vector spaces of different dimensions, highlighting their similarities, properties, and motivations.

## Contribution

It provides a comprehensive overview of three determinant generalisations, clarifying their formal relationships and properties, which aids understanding of interdimensional transformations.

## Key findings

- Identifies three main generalisations: determinant-like function, vector determinant, g-determinant.
- Discusses formal similarities and differences among these generalisations.
- Summarizes known properties and motivations for each generalisation.

## Abstract

Significant research has been carried out in the past half-century on defining generalised determinants for transformations between (typically real) vector spaces of different dimensions. We review three different generalisations of the determinant to non-square matrices, that we term for convenience the determinant-like function, the vector determinant and the g-determinant. We introduce and motivate these generalisations, note certain formal similarities between them and discuss their known properties.