Abstract definition
Let
be a field of characteristic zero, such as
or
.
Let
be a finite-dimensional vector space over
, and let
be an order-d simple tensor, i.e., there exist some vectors
such that
. If we are given a collection of linear maps
, then the multilinear multiplication of
with
is defined[1] as the action on
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. That is, for a general tensor
, the multilinear multiplication is
where
with
is one of
'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:
and
where
is the identity operator.
Definition in coordinates
In computational multilinear algebra it is conventional to work in coordinates. Assume that an inner product is fixed on
and let
denote the dual vector space of
. Let
be a basis for
, let
be the dual basis, and let
be a basis for
. The linear map
is then represented by the matrix
. Likewise, with respect to the standard tensor product basis
, the abstract tensor
is represented by the multidimensional array
. Observe that
where
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
becomes
The resulting tensor
lives in
.
Element-wise definition
From the above expression, an element-wise definition of the multilinear multiplication is obtained. Indeed, since
is a multidimensional array, it may be expressed as
where
are the coefficients. Then it follows from the above formulae that
where
is the Kronecker delta. Hence, if
, then
where the
are the elements of
as defined above.
Properties
Let
be an order-d tensor over the tensor product of
-vector spaces.
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]
It follows from the definition that the composition of two multilinear multiplications is also a multilinear multiplication:[1][2]
where
and
are linear maps.
Observe specifically that multilinear multiplications in different factors commute,
if
Computation
The factor-k multilinear multiplication
can be computed in coordinates as follows. Observe first that
Next, since
there is a bijective map, called the factor-k standard flattening,[1] denoted by
, that identifies
with an element from the latter space, namely
where
is the jth standard basis vector of
,
, and
is the factor-k flattening matrix of
whose columns are the factor-k vectors
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.