7.4 Tensor Component Array Representation

Tensor Component Array Representation is the concrete rendering of a tensor's components as a structured, multi-dimensional array of numbers, providing the practical form in which tensors are stored, manipulated, and computed with once an abstract multilinear object has been reduced to a fixed basis.

Definition and Scope

From Abstract Tensor to Array

Given a tensor of type ((p,q)) on an (n)-dimensional space and a chosen basis, the array representation collects every component

into a single object with (p+q) axes, each of length (n), so that the tensor becomes addressable as an ordinary multi-dimensional array rather than as an abstract map between vector spaces.

Correspondence to Familiar Structures

At low rank, the array representation reduces to structures already familiar from linear algebra: a rank-0 tensor is represented as a single scalar, a rank-1 tensor as a one-dimensional list of (n) numbers, and a rank-2 tensor as an (n \times n) matrix, with higher ranks extending the same pattern to cubes and hyper-cubes of numbers.

Structural Properties

Storage Layout

An array representation must specify how its multi-dimensional structure is laid out in a one-dimensional memory space, typically through a row-major or column-major convention. For a rank-2 array of dimension (n) stored row-major, the entry at position ((i,j)) resides at linear offset

with the generalization to higher rank obtained by nesting this rule one axis at a time.

Dense Versus Sparse Representation

When most components of a tensor are zero, as with tensors built from highly symmetric or localized structures, a dense array representation storing every entry becomes wasteful, motivating sparse representations that store only nonzero entries together with their index addresses. The choice between dense and sparse array representation is a computational decision layered on top of the same underlying component data.

Independence From, and Link To, the Abstract Tensor

The array representation is tied to the basis in which it was computed; the same tensor produces a different array under a different basis, following the standard component transformation law, so the array representation alone, without knowledge of the basis it belongs to, describes the tensor only partially.

Role Within Tensor Algebra

Computational Foundation

Every numerical tensor operation, addition, scalar multiplication, tensor product, contraction, and index permutation, is ultimately implemented as an operation on array representations, translating the abstract algebraic definition into loops, sums, and index manipulations over the stored entries.

Software and Library Design

Tensor libraries in scientific computing are built around array representations, exposing operations such as reshaping, slicing, and broadcasting that manipulate the array structure directly, while relying on the underlying algebraic theory of tensors to guarantee that these manipulations correspond to meaningful mathematical operations on the tensors they represent.