1.2.56 Tensor Component Array Definition

The Tensor Component Array Definition organizes multidimensional data using indexed components, essential for representing tensors in algebraic structures.

Tensor Component Array Definition is the specification of the full collection of numerical values that represent a tensor once a basis has been chosen, organized as a multidimensional array whose number of dimensions equals the tensor's rank and whose extent along each dimension equals the dimension of the underlying vector space. Where a single component is one scalar entry picked out by fixing every index, the component array is the entire structured collection of all such entries taken together, indexed systematically by every possible combination of index values.

Structure of the Array

Rank and Array Dimensionality

A tensor of type (p, q) has rank p + q, and its component array is accordingly an array with p + q dimensions, sometimes called an (p + q)-dimensional array or a way to distinguish it from the dimension n of the underlying vector space itself. Each of the p + q axes of the array corresponds to one index of the tensor, either upper or lower.

array dimensions = p + q

Extent Along Each Axis

Every axis of the component array has the same extent, equal to n, the dimension of the vector space V on which the tensor is defined. This uniform extent follows because every index, whether upper or lower, ranges over the same set of basis labels 1 through n.

Total Number of Entries

The total number of scalar entries in the component array is n raised to the power of the rank:

total entries = n^{p + q}

For example, a type (1, 1) tensor in a 3-dimensional space has 3^2 = 9 entries, naturally arranged as a 3 × 3 matrix, while a type (0, 3) tensor in the same space has 3^3 = 27 entries.

Low-Rank Array Examples

Rank Zero: A Single Entry

A type (0, 0) tensor has rank zero, and its component array degenerates to a single scalar entry, with no indices needed to select it.

Rank One: A List

A type (1, 0) or type (0, 1) tensor has rank one, and its component array is a one-dimensional list of n entries, indexed by a single upper or lower index respectively.

(\begin{matrix} T^{1} \\ T^{2} \\ ⋮ \\ T^{n} \end{matrix})

Rank Two: A Matrix

A type (1, 1), (2, 0), or (0, 2) tensor has rank two, and its component array is a two-dimensional grid of n^2 entries, naturally displayed as an n × n matrix, with rows and columns corresponding to the two indices.

(\begin{matrix} T_{1}^{1} & T_{1}^{2} \\ T_{2}^{1} & T_{2}^{2} \end{matrix})

Rank Three and Higher: Blocks of Matrices

A rank-three tensor's component array can be visualized as a stack of matrices, one matrix for each value of the third index, and a rank-four tensor as a grid of such stacks, and so on. Beyond rank two, the array is no longer conveniently displayed on a flat page and is instead described symbolically through its indexed components.

The Array Is Not the Tensor

Basis Dependence

The component array is tied to a specific choice of basis; changing the basis produces a different array of numbers even though it represents the same underlying abstract tensor. The array is therefore a representation of the tensor in coordinates, not the tensor itself, which exists independently of any basis.

Consistency Requirement

Not every array of numbers indexed in this fashion qualifies as the component array of a tensor. For the array to represent a genuine tensor, its entries must obey the prescribed transformation law relating the array computed in one basis to the array computed in any other basis; an array that fails to transform this way is merely a collection of numbers, not a tensor.

Diagrammatic Summary

The diagram shows a rank-two component array for a tensor built over a three-dimensional vector space, containing 3^2 = 9 entries arranged in a grid, each entry labeled by the two index values that select it.