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Function Repository Resource:

 HyperDet

Source Notebook

Compute the hyperdeterminant for a given hypermatrix (a multidimensional array of complex numbers)

Contributed by: Riccardo Gatti

ResourceFunction["HyperDet"][h]

gives the hyperdeterminant of the cubical hypermatrix (or hypercubical) h.

Details

A 2-dimensional n×n=n2 square matrix is a particular case of the more general d-dimensional n×…×n=nd cubical hypermatrix (d-hypermatrix), hence the hyperdeterminant is a more general form of the determinant. There are two ways to generalize the determinant to hyperdeterminant: the combinatorial way and the geometric way.
Let A be a d-hypermatrix. The combinatorial hyperdeterminant of A is defined as where Sn is the set symmetric group on n letters and sgn is the permutation signature function.
There is also a definition with a geometric interpretation.

Examples

Basic Examples (1) 

The hyperdeterminant of a cubical hypermatrix where d=4 and n=2:

In[1]:=
ResourceFunction["HyperDet"][ Array[Subscript[W, #1, #2, #3, #4] &, {2, 2, 2, 2}]] // Simplify
Out[1]=

Publisher

Riccardo Gatti

Version History

  • 1.0.0 – 18 November 2022

Source Metadata

  • Citation:
    • Lim, Lek-Heng. "Tensors and hypermatrices." Handbook of linear algebra (2013): 231-260. Link.

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