4.22.2 Tensor Multilinear Matrix Case

Tensor Multilinear Matrix Case is the arity-two instance of representing a multilinear map by an explicit array of numbers, the case in which the representing array is an ordinary two-dimensional matrix rather than a higher-dimensional hypermatrix. It serves as the base case and template for the general theory of representing multilinear maps of any arity, since matrices are the most familiar and best-understood instance before the pattern is extended to three or more argument slots.

The Matrix Case for Scalar Output

Bilinear Forms as Matrices

For a bilinear form f: V × W → F with scalar output, choosing bases {eᵢ} of V and {fⱼ} of W produces the matrix Aᵢⱼ = f(eᵢ, fⱼ), and every value of f is recovered as f(v,w) = v^T A w. This is the simplest and most complete instance of the matrix case: a single ordinary matrix fully captures the bilinear map.

The Matrix Case for Vector-Valued Bilinear Maps

Why a Single Matrix No Longer Suffices

When the codomain U of a bilinear map g: V × W → U has dimension greater than one, no single ordinary matrix can represent g completely, since g(v,w) must itself be a vector in U, not a scalar. The matrix case for vector-valued bilinear maps instead requires one matrix per coordinate of the output.

A Stack of Matrices

If U has basis {u₁,...,u_p}, then g is represented by p separate matrices A^{(1)}, ..., A^{(p)}, one for each output coordinate, with

where g^k denotes the k-th coordinate of g relative to the basis of U; each entry of the tuple is obtained the same way an ordinary bilinear form's matrix entry is obtained, but now separately for each output coordinate. The value g(v,w) is recovered coordinate by coordinate, k-th coordinate equal to v^T A^{(k)} w.

Reassembling Into a Single Object

The stack of p matrices A^{(1)},...,A^{(p)} is naturally organized as a single three-index array T_{kij} = g^k(eᵢ,fⱼ), the first genuine instance, as the arity or output dimension grows, of the higher-dimensional hypermatrices needed once the matrix case no longer applies in its simplest single-matrix form.

The Matrix Case as Template for Higher Arity

Extending to Trilinear Maps

For a trilinear form h: V₁ × V₂ × V₃ → F, the natural generalization of the matrix case is a three-index array T_{ijk} = h(e_i, f_j, g_k), which can be understood, and often is computed, as a stack of ordinary matrices, one matrix T_{··k} for each fixed value of the third index, reducing the trilinear case to a sequence of ordinary matrix cases glued together along one axis.

Matrix Slices of Higher-Arity Arrays

More generally, an n-index array representing an n-ary multilinear form can be examined one "matrix slice" at a time, by fixing all but two of its indices and reading off the resulting two-index sub-array as an ordinary matrix; this technique reduces questions about the structure of a high-arity multilinear map to a sequence of matrix-case questions applied to its slices.

Distinguishing the Matrix Case From the Operator Case

Bilinear Map Matrices Versus Linear Operator Matrices

The matrix representing a bilinear form transforms under change of basis by congruence, A ↦ P^T A Q, whereas the matrix representing a linear operator transforms by similarity, A ↦ P^{-1} A P; the matrix case for multilinear maps must not be confused with the ordinary matrix representation of a linear map between two vector spaces, since the two situations, though both producing an array called a "matrix," obey different transformation rules and arise from different kinds of underlying object.

The Matrix Case Within the Type (p,q) Classification

A bilinear form corresponds to a type (0,2) tensor, purely covariant, whose matrix case gives the familiar bilinear-form matrix; a linear operator corresponds to a type (1,1) tensor, mixed, whose "matrix case" is the familiar operator matrix; recognizing which type is in play clarifies which transformation law, congruence or similarity, governs the matrix obtained.

Computational Advantages of the Matrix Case

Leveraging Standard Linear Algebra

Because the matrix case reduces a bilinear form to an ordinary matrix, all standard numerical linear algebra, matrix multiplication, eigenvalue computation, singular value decomposition, LU or Cholesky factorization, becomes directly applicable to studying the bilinear form, without requiring any specialized multilinear-algebra software.

Rank and Structure Visible in the Matrix

Properties of a bilinear form that are otherwise abstractly defined, non-degeneracy, symmetry type, rank, become immediately visible and computable once the matrix case representation is in hand: invertibility of A for non-degeneracy, A = A^T for symmetry, and the rank of A computed via row reduction or singular value decomposition for the rank of the form.

When the Matrix Case Is Insufficient

Arity Beyond Two Requires Hypermatrices

Once arity exceeds two, no single matrix, in the ordinary two-index sense, can represent the full multilinear map; a genuine n-index hypermatrix is required, and although matrix slicing offers a way to inspect such an object using familiar matrix tools, no single slice, nor any finite collection of two-index matrices without additional bookkeeping, replaces the full hypermatrix as the complete and faithful representation of the multilinear map.