Generalisations of the determinant to interdimensional transformations: a review

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.