Defining the Determinant-like Function for m by n Matrices Using the Exterior Algebra

Abstract

In this paper, we present a function called the Determinant-Like Function that generalises the determinant function to m by n matrices using the Clifford algebra Cl(V, 0). By definition, this generalisation satisfies the property that the exterior product of its column vectors has a magnitude that equals its determinant-like function if it has more rows than columns. For matrices which have more columns than rows, we make use of another property that the exterior product of its rows has a magnitude that equals the absolute value of its determinant-like function if it has more columns than rows. Defining the sign of this determinant-like function remains an open question.