Multilinear multiplication

In multilinear algebra, applying a map that is the tensor product of linear maps to a tensor is called a multilinear multiplication.
Abstract definition.
Let formula_1 be a field of characteristic zero, such as formula_2 or formula_3.
Let formula_4 be a finite-dimensional vector space over formula_5, and let formula_6 be an order-d simple tensor, i.e., there exist some vectors formula_7 such that formula_8. If we are given a collection of linear maps formula_9, then the multilinear multiplication of formula_10 with formula_11 is defined as the action on formula_10 of the tensor product of these linear maps, namely
formula_13
Since the tensor product of linear maps is itself a linear map, and because every tensor admits a tensor rank decomposition, the above expression extends linearly to all tensors. That is, for a general tensor formula_6, the multilinear multiplication is
formula_15
where formula_16 with formula_17 is one of formula_10'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 formula_10 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:formula_20andformula_21where formula_22 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 formula_4 and let formula_24 denote the dual vector space of formula_25. Let formula_26 be a basis for formula_25, let formula_28 be the dual basis, and let formula_29 be a basis for formula_30. The linear map formula_31 is then represented by the matrix formula_32. Likewise, with respect to the standard tensor product basis formula_33, the abstract tensorformula_34is represented by the multidimensional array formula_35 . Observe that formula_36
where formula_37 is the "j"th standard basis vector of formula_38 and the tensor product of vectors is the affine Segre map formula_39. It follows from the above choices of bases that the multilinear multiplication formula_40 becomes
formula_41
The resulting tensor formula_42 lives in formula_43.
Element-wise definition.
From the above expression, an element-wise definition of the multilinear multiplication is obtained. Indeed, since formula_42 is a multidimensional array, it may be expressed as formula_45where formula_46 are the coefficients. Then it follows from the above formulae that
formula_47
where formula_48 is the Kronecker delta. Hence, if formula_49, then
formula_50
where the formula_51 are the elements of formula_52 as defined above.
Properties.
Let formula_53 be an order-d tensor over the tensor product of formula_54-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):
formula_55 
Multilinear multiplication is a linear map: formula_56 
It follows from the definition that the composition of two multilinear multiplications is also a multilinear multiplication:
formula_57
where formula_58 and formula_59 are linear maps.
Observe specifically that multilinear multiplications in different factors commute,
formula_60
if formula_61
Computation.
The factor-k multilinear multiplication formula_62 can be computed in coordinates as follows. Observe first that
formula_63
Next, since
formula_64
there is a bijective map, called the factor-"k" standard flattening, denoted by formula_65, that identifies formula_66 with an element from the latter space, namely
formula_67
where formula_68is the "j"th standard basis vector of formula_69, formula_70, and formula_71 is the factor-"k" flattening matrix of formula_10 whose columns are the factor-"k" vectors formula_73 in some order, determined by the particular choice of the bijective map
formula_74
In other words, the multilinear multiplication formula_75 can be computed as a sequence of "d" factor-"k" multilinear multiplications, which themselves can be implemented efficiently as classic matrix multiplications.
Applications.
The higher-order singular value decomposition (HOSVD) factorizes a tensor given in coordinates formula_76 as the multilinear multiplication formula_77, where formula_78 are orthogonal matrices and formula_79.