1.5.5 Component Tensor Representation

Component Tensor Representation breaks down complex tensors into coordinate-dependent components, essential for calculations in physics and engineering.

Component Tensor Representation is the description of a tensor by the ordered array of numbers obtained once a basis of the underlying vector space, and the corresponding dual basis of its dual space, have been fixed. Where a tensor exists as an abstract multilinear object independent of any basis, its component representation is the concrete numerical encoding of that object relative to a chosen frame, consisting of one number for every combination of index values the tensor's upper and lower indices can take.

Definition and Basic Structure

Components Relative to a Basis

Given a basis e_1, ..., e_n of an n-dimensional vector space V and the corresponding dual basis e^1, ..., e^n of V*, a tensor T of type (p, q) is represented by the collection of numbers obtained by evaluating T on all combinations of basis covectors and basis vectors placed in its argument slots.

T_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}} = T (e^{1} \dots e^{p} e_{1} \dots e_{q})

Number of Components

A tensor of type (p, q) over an n-dimensional vector space has n raised to the power p + q components, since each of the p + q indices independently ranges over n values. A scalar, with p = q = 0, has a single component; a vector or covector, with p + q = 1, has n components; a rank-two tensor has n^2 components.

number of components = n^{p + q}

Index Placement and Its Meaning

Upper Indices for Contravariant Slots

Each upper index of a component array corresponds to an argument slot of the tensor that accepts a covector, and the values taken by an upper index label the coefficients of the tensor's contravariant behavior, meaning the way those components respond to a change of basis mirrors the way vector components respond.

Lower Indices for Covariant Slots

Each lower index corresponds to an argument slot that accepts a vector, and the values taken by a lower index label the coefficients of the tensor's covariant behavior, meaning those components respond to a change of basis in the same manner as covector components.

Ordering of Indices

The relative order of upper indices among themselves, and of lower indices among themselves, is part of the definition of the component array, since permuting indices can change which components are equal to which for tensors that are not fully symmetric or fully antisymmetric.

Reconstructing the Tensor from Its Components

The Expansion Formula

The abstract tensor is recovered from its components by summing the components against the tensor products of the corresponding basis vectors and dual basis covectors, weighted appropriately, using the Einstein summation convention to suppress the explicit summation symbols.

T = T_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}} e_{i_{1}} \otimes \dots \otimes e_{i_{p}} \otimes e^{j_{1}} \otimes \dots \otimes e^{j_{q}}

Basis Dependence of the Numbers, Basis Independence of the Object

The array of numbers changes when the basis changes, but the sum described above always reconstructs the same abstract tensor T, regardless of which basis was used. This is the precise sense in which the components are a representation of the tensor rather than the tensor itself.

Component Behavior Under a Change of Basis

The Transformation Law

If A denotes the matrix expressing a new basis in terms of the old one and A^{-1} its inverse, then the components of a type (p, q) tensor in the new basis are obtained by contracting the old components with one factor of A for each upper index and one factor of A^{-1} for each lower index.

T_{' l_{1}}^{' k^{1}} = A_{i}^{k} B_{l}^{j} T_{j}^{i}

Consistency Requirement

Not every array of numbers indexed by p + q indices qualifies as the components of a genuine tensor; the array must transform according to this law under every admissible change of basis for it to represent a well-defined, basis-independent multilinear object.

Practical Role of Component Representation

Computation

Component representation is what makes tensor operations computable: addition, tensor product, contraction, and index raising and lowering are all carried out as explicit arithmetic on arrays of numbers once a basis has been fixed, even though the operations themselves have basis-independent meanings.

Choice of Convenient Bases

Because the component array depends on the chosen basis, a basis can be selected to make a particular tensor's components as simple as possible, such as an orthonormal basis that diagonalizes a symmetric tensor, without altering the underlying tensor being described.

Diagrammatic Summary

The diagram traces the two-way relationship of component tensor representation: fixing a basis turns the abstract tensor T into a concrete array of numbers, and changing the basis transforms that array according to a fixed law while leaving the abstract tensor it represents unchanged.