9.11.4 Tensor Coordinate Array Construction

Tensor Coordinate Array Construction is the step of the coordinate representation process in which every individually assigned component of a tensor is assembled into a single, complete multi-dimensional array, organized so that each entry occupies the position dictated by its own index address, producing one unified data structure that stands for the tensor as a whole rather than a scattered collection of separately computed numbers; it is the assembly stage that follows component assignment and precedes any further use of the tensor's components as a whole.

How the Array Is Assembled

Iterating Over Every Admissible Index Combination

Array construction proceeds by considering every combination of index values drawn from the coordinate label set, once for each free index the tensor carries, and placing the corresponding assigned component into the array position matching that combination.

Completing the Array Requires Every Position to Be Filled

The construction of a tensor's coordinate array is not finished until every admissible index combination has been assigned its entry; an array with any position left unfilled does not yet fully represent the tensor, since any subsequent operation might depend on the missing entry.

The Shape of the Constructed Array

Dimensionality Matches the Tensor's Order

A tensor of order k is constructed into an array with k independent index positions, each ranging over the coordinate label set, so the array's dimensionality as a data structure directly mirrors the number of slots the tensor itself has.

Size Along Each Position Matches the Space's Dimension

Because each index ranges over the same coordinate label set, every dimension of the constructed array has the same size, equal to the dimension of the underlying space, giving the array a uniform, regular shape regardless of which particular index position is being considered.

Why Construction Is Treated as a Distinct Step

Component Assignment Alone Produces Only Individual Values

The mechanism of component assignment produces one numerical value at a time, for one index combination; array construction is the additional step required to gather all such individually produced values into a single organized structure that can be handled, referenced, or passed along as a whole.

A Constructed Array Enables Whole-Object Operations

Once assembled, the array as a whole becomes the object that further operations act upon — printed, compared, transformed, or stored — rather than requiring each entry to be separately tracked and referenced by its individual index combination every time the tensor's components are needed.

Diagram of Array Construction

Consequences of Array Construction

It Provides a Single Object Fit for Later Manipulation

By producing one assembled array in place of many separately assigned components, array construction gives later stages of a calculation, such as tensor addition or contraction, a single well-defined object to operate on, avoiding the need to re-derive or re-collect individual entries at every subsequent step.

An Incomplete Construction Undermines Everything Built Upon It

Because operations performed on a constructed array assume that every position has been correctly filled, any gap or mistake introduced during construction propagates directly into every later calculation relying on that array, making the completeness and correctness of construction a prerequisite for trusting any of the tensor's subsequent processing.