5.23 Tensor Product Component Representation

Tensor Product Component Representation is the description of an element of a tensor product V ⊗ W by its array of scalar coefficients relative to chosen bases of V and W, converting the abstract, basis-free tensor product into a concrete object of coordinates that can be tabulated, computed with, and related to matrices and multi-dimensional arrays. Component representation is the standard bridge between the abstract algebraic theory of tensor products and the explicit, numerical practice of working with tensors in linear algebra, physics, and engineering.

Defining the Component Representation

Coordinates Relative to a Basis

Given bases {eᵢ}_{i=1}^m of V and {fⱼ}_{j=1}^n of W, the family {eᵢ ⊗ fⱼ} is a basis of V ⊗ W, so every element t ∈ V ⊗ W has a unique expansion

and the scalars c_{ij} are called the components of t relative to the chosen bases.

Arranging Components into a Matrix

Because the indices i and j each range over a finite set, the components c_{ij} can be arranged into an m × n matrix C, giving every element of V ⊗ W a concrete matrix representation once bases are fixed, identical in form to the matrix representation of a linear map W* → V or a bilinear form on V* × W*.

Component Representation of Simple versus General Tensors

Simple Tensors as Rank-One Component Arrays

If t = v ⊗ w is a simple tensor with v = Σ aᵢeᵢ and w = Σ bⱼfⱼ, expanding gives c_{ij} = aᵢ bⱼ, so the component matrix of a simple tensor is precisely the outer product C = ab^T, a rank-one matrix; this is the component-level signature distinguishing simple tensors from general ones.

General Tensors and Matrix Rank

A general tensor's component matrix C need not be rank one; the minimal number of simple tensors needed to express t as a sum equals the matrix rank of C, connecting the abstract notion of tensor rank directly to the ordinary linear-algebraic rank of the component representation.

Diagram of the Component Grid

Change of Basis

Transformation Rule for Components

If the bases {eᵢ} and {fⱼ} are replaced by new bases {e′ᵢ} and {f′ⱼ} related by change-of-basis matrices P and Q (so eᵢ = Σ P_{ki} e′ₖ and fⱼ = Σ Q_{lj} f′ₗ), the component array of a fixed tensor t transforms as

or in matrix form C′ = P C Qᵀ, generalizing the transformation rule for matrices representing bilinear forms and confirming that the tensor t itself is basis-independent even though its component array changes with the choice of basis.

Covariance and Contravariance in Notation

In classical tensor notation, distinguishing indices that transform "with" the basis change (covariant) from those that transform "against" it (contravariant) refines the component representation further, tracking whether a given tensor factor comes from V or from the dual space V*, and dictating precisely how each index in the change-of-basis formula must be transformed.

Extension to Higher-Order Tensors

Multi-Index Component Arrays

For a tensor product of n spaces, the component representation generalizes to an n-index array c_{i₁ i₂ ... iₙ}, one index per tensor factor, with the same principle that fixing bases for each factor produces a unique multi-dimensional array of coefficients representing any given element of the tensor product.

Practical Role in Computation

Multi-index component arrays are the data structures underlying numerical tensor computations throughout applied mathematics, physics, and machine learning, where operations such as contraction, tensor products of maps, and symmetrization are all implemented as explicit operations on these component arrays.

Significance of Component Representation

Making the Abstract Tensor Product Computable

Component representation is what transforms the basis-free, universal-property definition of the tensor product into an object suitable for explicit calculation, giving every abstract tensor a concrete numerical form once bases are fixed, without which most practical applications of tensor products would be inaccessible.

Preserving Basis-Independence of the Underlying Object

Despite depending on a choice of basis for its explicit form, component representation always transforms correctly under change of basis, guaranteeing that the tensor itself — as opposed to its component array — remains the well-defined, basis-independent object that the abstract tensor product theory guarantees it to be.