14.18.4 Tensor Map Product Basis Ordering Convention

The Tensor Map Product Basis Ordering Convention defines a standard for ordering basis elements in tensor product spaces to ensure consistent algebraic computations.

Tensor Map Product Basis Ordering Convention is the choice of how the doubly indexed family of basis tensors $e_{i} \otimes f_{j}$ is arranged into a single linearly ordered list, a choice that fixes which row or column of the matrix representing $f \otimes g$ corresponds to which pair of indices, without affecting the underlying linear map itself.

The Need for an Ordering

Two Indices, One List

A basis tensor of $V \otimes W$ is naturally labeled by a pair of indices $(i, j)$ , with $1 \leq i \leq n$ and $1 \leq j \leq m$ , but writing the components of a vector or the entries of a matrix requires a single linear index running from $1$ to $n m$ . The ordering convention is the rule that converts the pair $(i, j)$ into that single index.

The Standard Lexicographic Rule

The most common convention orders pairs lexicographically by $i$ first, then by $j$ , assigning the linear index

k = (i - 1) m + j

to the pair $(i, j)$ , so that $j$ varies fastest as $k$ increases, and $i$ only increases once $j$ has cycled through all its values; this is the ordering under which the matrix of $f \otimes g$ equals the Kronecker product $F \otimes G$ in its standard block form.

Effect on the Matrix Presentation

Block Structure Under the Standard Convention

Under the standard convention, the matrix $F \otimes G$ is arranged in $n$ by $n$ blocks, with the block in position $(i, a)$ equal to $F_{i}^{a} G$ , that is, the matrix $G$ scaled by the single scalar entry $F_{i}^{a}$ . Reversing the roles, ordering by $j$ first and $i$ second, produces instead a block structure with the block in position $(j, b)$ equal to $G_{j}^{b} F$ , which is the matrix $G \otimes F$ rather than $F \otimes G$ .

Permutation Relating Different Conventions

The two resulting matrices are related by conjugation with a fixed permutation matrix $P$ that reorders the basis, since both list the same set of basis tensors, only in a different sequence,

G \otimes F = P (F \otimes G) P^{- 1}

with $P$ known explicitly as the commutation matrix associated with the dimensions $n$ and $m$ , so that switching ordering conventions never changes the map, only the labeling scheme by which its matrix entries are addressed.

What the Convention Does Not Affect

Invariance of the Map and Its Entries as a Function

The value $(f \otimes g) (e_{i} \otimes f_{j})$ assigned to a basis tensor by the tensor map product basis formula does not depend on how the pair $(i, j)$ is later encoded as a linear index: the formula refers to $e_{i}$ and $f_{j}$ directly, not to their position in some flattened list. The ordering convention only governs how that same information is displayed once it is written as a matrix.

Consequence for Bookkeeping, Not for Content

Two practitioners using different ordering conventions compute the same coefficients for the same basis tensors; they only disagree about which row or column of their respective matrices those coefficients occupy. Any formula, theorem, or computation stated in terms of the basis tensors themselves, rather than in terms of the position of an entry inside a flattened matrix, is therefore completely unaffected by the choice of ordering convention.

Practical Considerations

Consistency With Iterated Tensor Products

When more than two maps are tensored together, for example $f \otimes g \otimes h$ , the ordering convention must be extended consistently across all three index sets simultaneously, typically by nesting the same lexicographic rule, first grouping the last two indices together and then combining the result with the first, matching the standard associativity convention used to identify $(V \otimes W) \otimes U$ with $V \otimes (W \otimes U)$ .

Interaction With Software Implementations

Numerical libraries that implement the Kronecker product fix one specific ordering convention internally, generally the lexicographic one with the second factor varying fastest, and any formula transcribed from an abstract index calculation into such software must respect that fixed convention or the resulting matrix will represent a permuted, rather than literally equal, version of the intended map.