A Novel Hypermatrix Product and its Application to Multilinear Mappings

Abstract

Matrix theory provides a well-established algebraic framework for working with linear maps, in which matrix multiplication replaces the composition of linear transformations. However, there is no canonical multiplication rule for hypermatrices that leads to multilinear maps, partly because multilinear maps are not closed under composition. To address this gap, this research introduces a novel (restricted) hypermatrix multiplication based on the Frobenius inner product. We start byshowing that every multilinear map 𝑓: 𝑉1 × 𝑉2 × … × 𝑉𝑛 → 𝑉0 gives a hypermatrix representation 𝒜 and defining a contraction operation, which computes 𝑓(𝑣1, 𝑣2, … , 𝑣𝑛 ) through Frobenius inner products between 𝒜 and matrices derived from input vectors. This operation allows for the efficient computation of the hypermatrix of an arbitrary multilinear map. This work provides constructive proofs and detailed numerical examples.