4.18.3 Tensor Multilinear Form Component Array

A tensor multilinear form component array organizes coefficients of multilinear maps acting on basis vectors for structured algebraic computation.

Tensor Multilinear Form Component Array is the finite table of scalars obtained by evaluating a multilinear form on every tuple of basis vectors drawn from chosen bases of its argument spaces, indexed by one index per argument slot. The component array is the concrete, computable object through which an abstract multilinear form is stored, manipulated, and reconstructed once bases have been fixed.

Construction of the Array

Evaluating on Basis Tuples

Given a multilinear form f: V₁ × ... × Vₙ → F and bases {e^{(1)}_1, ..., e^{(1)}_{d₁}}, ..., {e^{(n)}_1, ..., e^{(n)}_{dₙ}} for the respective spaces, the component array is defined entrywise by

T_{k_{1} \dots k_{n}} = f (e_{k_{1}}^{(1)}, \dots, e_{k_{n}}^{(n)})

with each index kᵢ ranging from 1 to dᵢ = dim(Vᵢ). The array has one entry for every choice of one basis vector from each of the n factors, and so has d₁ × d₂ × ... × dₙ entries in total.

Size and Shape

The shape of the array mirrors the arity and the dimensions of the factor spaces: a linear form (n = 1) has a component array that is a single list of d₁ numbers, a bilinear form (n = 2) has a d₁ × d₂ matrix, and an n-ary form has an n-dimensional array, sometimes called a hypermatrix, with dᵢ entries along its i-th axis.

Reconstructing the Form From Its Array

The Expansion Formula

Any vector vᵢ ∈ Vᵢ decomposes in its basis as vᵢ = ∑ₖ vᵢᵏ e^{(i)}_k for coordinates vᵢᵏ ∈ F. Multilinearity of f, applied one slot at a time, gives the reconstruction formula

f (v_{1}, \dots, v_{n}) = \sum k_{1} \dots \sum k_{n} v_{1}^{k_{1}} \dots v_{n}^{k_{n}} T_{k_{1} \dots k_{n}}

so the entire form is recovered from the array by summing products of input coordinates against array entries. No information about f is lost by passing to its component array, and none needs to be added beyond the array to recover f fully.

Well-Definedness Requires No Extra Data

The reconstruction formula shows that the component array, together with the choice of bases used to build it, is a complete description of f: any two multilinear forms with the same component array relative to the same bases are equal as maps, since they agree on all inputs by the expansion formula above, and any two forms differing on some input must differ in at least one array entry.

Special Cases

Vectors as Components of Linear Forms

For n = 1, the component array reduces to a list T_k = f(e_k), and the reconstruction formula reduces to f(v) = ∑ₖ v^k T_k, matching the familiar description of a linear functional as a row of coefficients dotted against a vector of coordinates.

Matrices as Components of Bilinear Forms

For n = 2, T_{k₁k₂} = f(e^{(1)}_{k₁}, e^{(2)}_{k₂}) is precisely the matrix representing the bilinear form, and the reconstruction formula becomes the familiar f(v, w) = v^T T w written in summation form, f(v,w) = ∑_{k₁,k₂} v^{k₁} w^{k₂} T_{k₁k₂}.

Higher Arrays for Trilinear and Beyond

For n = 3, the component array is a three-index object T_{k₁k₂k₃}, which can be pictured as a stack of matrices, one for each value of k₁, or equivalently sliced along any one of its three axes; the same reconstruction formula applies with three coordinate sums instead of two.

Operations on Component Arrays

Contraction Along One Index

Fixing the vector in one slot to a basis vector e^{(i)}_{k} and letting the others vary corresponds, at the level of arrays, to fixing the i-th index of T to k and reading off the remaining lower-dimensional array as the component array of the resulting (n-1)-ary form.

Symmetrization and Antisymmetrization

When all factors coincide, the component array of the symmetrized form is obtained by averaging T over all permutations of its indices, T^{sym}_{k₁...kₙ} = (1/n!) ∑_σ T_{k_{σ(1)}...k_{σ(n)}}, and the antisymmetrized form's array is obtained the same way but with each term signed by the parity of σ; both operations act purely on the index pattern of the array without reference to the original vector spaces.

Combining Arrays via the Tensor Product of Forms

If T is the component array of a form f of arity m and S is the component array of a form g of arity k, the component array of the combined arity-(m+k) form (u, w) ↦ f(u)g(w) is the outer product array (T ⊗ S)_{k₁...kₘ,l₁...l_k} = T_{k₁...kₘ} S_{l₁...l_k}, obtained by multiplying every entry of T with every entry of S.

Practical Role of the Component Array

Bridge Between Abstract Theory and Computation

The component array is what allows multilinear forms, and by extension the tensors that are dual to them, to be entered into computer memory as finite blocks of numbers, manipulated with array-indexing operations, and processed by numerical linear algebra routines built for vectors, matrices, and higher-dimensional arrays, while the abstract theory of multilinear forms guarantees these array manipulations correspond faithfully to genuine operations on the underlying basis-free objects.

Cost of Representation

The number of entries in the component array, d₁ × ... × dₙ, grows multiplicatively with arity, so representing a general multilinear form of high arity on high-dimensional spaces requires storage growing exponentially in the number of argument slots; this cost is a primary motivation for identifying and exploiting additional structure, such as symmetry, sparsity, or low tensor rank, whenever a multilinear form of high arity must be stored or computed with directly.