Hyperdeterminant

Article summary

In algebra, the hyperdeterminant is a generalization of the determinant. Whereas a determinant is a scalar valued function defined on an n × n square matrix, a hyperdeterminant is defined on a multidimensional array of numbers or tensor. Like a determinant, the hyperdeterminant is a homogeneous polynomial with integer coefficients in the components of the tensor. Many other properties of determinants generalize in some way to hyperdeterminants, but unlike a determinant, the hyperdeterminant does not have a simple geometric interpretation in terms of volumes. There are at least three definitions of hyperdeterminant. The first was discovered by Arthur Cayley in 1843 presented to the Cambridge Philosophical Society. It is in two parts and Cayley's first hyperdeterminant is covered in the second part. It is usually denoted by det 0 . The second Cayley hyperdeterminant originated in 1845 and is often denoted "Det". This definition is a discriminant for a singular point on a scalar valued multilinear map. Cayley's first hyperdeterminant is defined only for hypercubes having an even number of dimensions (although variations exist in odd dimensions). Cayley's second hyperdeterminant is defined for a restricted range of hypermatrix formats (including the hypercubes of any dimensions).…

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