12.2.3 Tensor Composition Operation Area

The Tensor Composition Operation Area explores how tensors combine through operations, defining their algebraic structure and enabling complex mathematical modeling.

Tensor Composition Operation Area is the subset of tensor algebra concerned with building new tensors by combining the tensor product with contraction in a single coordinated step, joining two tensors together and simultaneously summing over one or more matched index pairs, the pattern underlying operations such as applying a linear map to a vector or multiplying two mixed tensors together.

Foundational Setting

Composition as Product Followed by Contraction

While the tensor product alone combines two tensors without reducing any indices, and contraction alone reduces the indices of a single tensor, composition names the frequently occurring combined pattern of forming a tensor product first and then immediately contracting one or more index pairs across the two original factors.

Why Composition Deserves Its Own Area

This combined pattern recurs so often, in applying linear maps, composing bilinear forms, and building chains of mixed tensors, that recognizing it as a coordinated single procedure, rather than two entirely separate steps, clarifies both its scope requirements and its effect on variance type.

The Composition Procedure

Step One: Form the Tensor Product

Given two tensors $S$ and $T$ , first form their tensor product, combining all of their indices into a single object:

P_{j l}^{i k} = S_{j}^{i} T_{l}^{k}

Step Two: Contract a Chosen Index Pair

Then contract one upper index from one factor against one lower index from the other, summing over that shared index letter:

R_{l}^{i} = \sum_{k} S_{k}^{i} T_{l}^{k}

Composition as Applying a Linear Map

The Simplest Instance

The composition of a mixed tensor with a vector, contracting the tensor's lower index against the vector's upper index, is exactly the operation of applying a linear map to that vector:

w^{i} = \sum_{j} T_{j}^{i} v^{j}

Composition of Two Linear Maps

Composing two mixed tensors representing linear maps, contracting the lower index of one against the upper index of the other, produces a third mixed tensor representing the composite linear map, mirroring ordinary matrix multiplication:

C_{k}^{i} = \sum_{j} A_{j}^{i} B_{k}^{j}

Visual Overview

Diagram of the Composition Pattern

Effect on Variance Type

Combining the Individual Effects

Since composition is the tensor product followed by contraction, its effect on variance type combines the two individual operations' effects directly: types add under the product, then one contravariant and one covariant count each decrease by one under the contraction, so a composition of tensors of type $(p_{1}, q_{1})$ and $(p_{2}, q_{2})$ yields type $(p_{1} + p_{2} - 1, q_{1} + q_{2} - 1)$ .

Associativity of Chained Composition

Because composition reduces to a sequence of well-defined product and contraction steps, composing three or more tensors in a chain gives a result independent of the order in which the individual compositions are carried out, mirroring the associativity of ordinary matrix multiplication.

Summary of Key Traits

Defining Characteristics

Composition combines the tensor product and contraction into a single coordinated procedure, joining two tensors while simultaneously summing over a matched index pair.
Applying a linear map to a vector, and composing two linear maps, are the most direct and familiar instances of this composition pattern.
The resulting variance type follows directly from combining the additive effect of the product with the subtractive effect of the contraction.
Chained compositions of several tensors are associative, mirroring the associativity of ordinary matrix multiplication.