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.
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Pyle H.: Non-Square Determinants and Multilinear Vectors. Mathematics Association of America. 35(2), 65–69 (1962)
Aslaksen H: Quarternionic Determinants. The Mathematical Intelligencer. 18(3), 57–65 (1996)
Kyrchei I.I.: Cramer’s rule for quarternionic systems of linear equations. Journal of Mathematical Sciences. 155(6), 839–858 (2008)
Radic M.: A Definition of Determinant of Rectangular Matrix. Glas. Mat. 1(21), 321–349 (1966)
The author would like to thank Dr Bharath Kalyan, National University of Singapore, for his valuable comments on earlier drafts.
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Abhimanyu, P.S. Defining the Determinant-like Function for m by n Matrices Using the Exterior Algebra. Adv. Appl. Clifford Algebras 23, 787–792 (2013). https://doi.org/10.1007/s00006-013-0416-2
- linear algebra
- clifford algebra
- exterior algebra