In multilinear algebra, applying a map that is the tensor product of linear maps to a tensor is called a multilinear multiplication.
be a finite-dimensional vector space over
be an order-d simple tensor, i.e., there exist some vectors
If we are given a collection of linear maps
of the tensor product of these linear maps,[2] namely
Since the tensor product of linear maps is itself a linear map,[2] and because every tensor admits a tensor rank decomposition,[1] the above expression extends linearly to all tensors.
's tensor rank decompositions.
The validity of the above expression is not limited to a tensor rank decomposition; in fact, it is valid for any expression of
as a linear combination of pure tensors, which follows from the universal property of the tensor product.
It is standard to use the following shorthand notations in the literature for multilinear multiplications:
In computational multilinear algebra it is conventional to work in coordinates.
Assume that an inner product is fixed on
denote the dual vector space of
Likewise, with respect to the standard tensor product basis
is the jth standard basis vector of
and the tensor product of vectors is the affine Segre map
It follows from the above choices of bases that the multilinear multiplication
From the above expression, an element-wise definition of the multilinear multiplication is obtained.
is a multidimensional array, it may be expressed as
Since a multilinear multiplication is the tensor product of linear maps, we have the following multilinearity property (in the construction of the map):[1][2]
Multilinear multiplication is a linear map:[1][2]
Observe specifically that multilinear multiplications in different factors commute,
The factor-k multilinear multiplication
there is a bijective map, called the factor-k standard flattening,[1] denoted by
is the jth standard basis vector of
is the factor-k flattening matrix of
in some order, determined by the particular choice of the bijective map
In other words, the multilinear multiplication
can be computed as a sequence of d factor-k multilinear multiplications, which themselves can be implemented efficiently as classic matrix multiplications.
The higher-order singular value decomposition (HOSVD) factorizes a tensor given in coordinates