Multiplicative Properties
One of the most familiar properties of determinants is the multiplication rule which is sometimes known as the Binet-Cauchy formula. For square n × n matrices A and B the rule says that
- det(AB) = det(A) det(B)
This is one of the harder rules to generalize from determinants to hyperdeterminants because generalizations of products of hypermatrices can give hypermatrices of different sizes. The full domain of cases in which the product rule can be generalized is still a subject of research. However there are some basic instances that can be stated.
Given a multilinear form f(x1, ..., xr) we can apply a linear transformation on the last argument using an n × n matrix B, yr = B xr. This generates a new multilinear form of the same format,
- g(x1,...,xr) = f(x1,...,yr)
In terms of hypermatrices this defines a product which can be written g = f.B
It is then possible to use the definition of the hyperdeterminant to show that
- det(f.B) = det(f) det(B)N/n
where n is the degree of the hyperdeterminant. This generalises the product rule for matrices.
Further generalizations of the product rule have been demonstrated for appropriate products of hypermatrices of boundary format
Read more about this topic: Hyperdeterminant, Properties of Hyperdeterminants
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