The second Cayley hyperdeterminant originated in 1845 and is often called " Det . " This definition is a discriminant for a singular point on a scalar valued multilinear map.
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I wonder if there's an elementary proof that a multilinear map with multiple arguments cannot be simultaneously linear in all of them unless it's the zero map, as stated in the article.
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The notion of "'alternatization "'( or "'alternatisation "'in British English ) is used to derive an alternating multilinear map from any multilinear map with all arguments belonging to the same space.
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The notion of "'alternatization "'( or "'alternatisation "'in British English ) is used to derive an alternating multilinear map from any multilinear map with all arguments belonging to the same space.
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The set of all such multilinear maps forms a vector space, called the tensor product space of type \ left ( r, \, s \ right ) at p and denoted by \ left ( T _ p \ right ) ^ r _ sM.
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In viewing a tensor as a multilinear map, it is conventional to identify the vector space " V " with the space of linear functionals on the dual of " V ", the double dual " V " " ".
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At p, these two vector spaces may be used to construct type \ left ( r, \, s \ right ) tensors, which are real-valued multilinear maps acting on the direct sum of r copies of the cotangent space with s copies of the tangent space.
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In the general case a hyperdeterminant is defined as a discriminant for a multilinear map " f " from finite-dimensional vector spaces " V " i to their underlying field " K " which may be \ mathbb { R } or \ mathbb { C }.
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More generally, a tensor of order " m " which takes in " n " vectors ( where " n " is between 0 and " m " inclusive ) will return a tensor of order, see Tensor : As multilinear maps for further generalizations and details.
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