5.23.2 Tensor Product Component Multiplication Rule

The Tensor Product Component Multiplication Rule defines how components of tensors combine under multiplication, essential for understanding tensor algebra operations.

Tensor Product Component Multiplication Rule is the formula describing how the components of the tensor product of two tensors are obtained by multiplying the components of the two original tensors, (s ⊗ t)_{(i,k),(j,l)} = s_{ij} t_{kl}, generalizing the familiar fact that the components of a simple tensor v ⊗ w are the pairwise products aᵢbⱼ of the components of v and w. This rule is the component-level content of the tensor product operation itself, converting the abstract construction ⊗ into the concrete, purely multiplicative procedure of forming all pairwise products of two component arrays.

The Rule for Vectors: A Foundational Case

Outer Product of Components

If v = Σ aᵢeᵢ ∈ V and w = Σ bⱼfⱼ ∈ W, the component multiplication rule states that the components of the simple tensor v ⊗ w, relative to the basis {eᵢ ⊗ fⱼ}, are the products

c = a_{i} b_{j}

recovering the outer product of the coordinate vectors a = (aᵢ) and b = (bⱼ), and confirming directly that simple tensors correspond to rank-one component matrices.

The General Rule for Two Tensors

Setup: Tensoring Two Already-Composite Tensors

Suppose s ∈ V ⊗ W has components s_{ij} (relative to bases {eᵢ} of V, {fⱼ} of W) and t ∈ X ⊗ Y has components t_{kl} (relative to bases {gₖ} of X, {hₗ} of Y). The tensor product s ⊗ t ∈ (V ⊗ W) ⊗ (X ⊗ Y) has components, relative to the basis {(eᵢ ⊗ fⱼ) ⊗ (gₖ ⊗ hₗ)}, given exactly by the products

{(s \otimes t)}_{i j k l} = s_{i j} t_{k l}

Derivation from Bilinearity

This rule follows directly from expanding s = Σ s_{ij}(eᵢ ⊗ fⱼ) and t = Σ t_{kl}(gₖ ⊗ hₗ) and distributing ⊗ bilinearly over both sums:

s \otimes t = \sum_{i, j, k, l} s_{i j} t_{k l} ((e_{i} \otimes f_{j}) \otimes (g_{k} \otimes h_{l}))

so no property of the tensor product beyond bilinearity is needed to justify the multiplication rule.

Diagram of Component Multiplication

Matrix Form: The Kronecker Product

Reduction to Matrix Kronecker Product

Arranging s's components into a matrix S and t's components into a matrix T, the component multiplication rule states that s ⊗ t corresponds exactly to the Kronecker product matrix S ⊗ T, whose (i,k),(j,l) block entry is s_{ij} t_{kl}, giving the abstract multiplication rule a standard, directly computable matrix form used throughout numerical linear algebra.

Consistency with Repeated Application

Applying the rule iteratively to tensor products of more than two tensors gives a component multiplication rule for any number of factors: the components of t₁ ⊗ t₂ ⊗ ... ⊗ tₙ are the product (t₁)_{i₁} (t₂)_{i₂} ⋯ (tₙ)_{iₙ} of the individual components, generalizing the two-factor rule term by term.

Special Cases

Multiplication by a Scalar Tensor

If t is a scalar (an element of the ground field, viewed as a 0-tensor), the rule reduces to ordinary scalar multiplication of s's components, since a scalar has a single "component" and the rule multiplies every component of s by it.

Multiplying Simple Tensors

If both s = v ⊗ w and t = x ⊗ y are simple, the rule confirms s ⊗ t = v ⊗ w ⊗ x ⊗ y is again simple, with components aᵢbⱼckdl (writing v, w, x, y in components aᵢ, bⱼ, cₖ, dₗ), consistent with the fact that a tensor product of simple tensors remains simple.

Significance of the Component Multiplication Rule

The Concrete Face of an Abstract Bilinear Operation

The component multiplication rule is what makes the tensor product operation directly computable: once components are known for two tensors, forming their tensor product requires nothing more than computing every pairwise product of components, without any reference to the more abstract universal-property definition.

Basis for Numerical Tensor Algorithms

Because the rule reduces tensor product formation to elementary multiplication of numbers (or, in matrix form, to the Kronecker product), it underlies the implementation of tensor product operations in numerical software, where large multi-dimensional tensors are routinely combined using exactly this pairwise multiplication procedure.