14.9.3 Tensor Kronecker Basis Ordering

Tensor Kronecker Basis Ordering is the convention that fixes how the basis vectors of two factor spaces are enumerated and paired when a Kronecker product is used to build the basis of the resulting composite space. It determines the precise sequence in which pairs of basis vectors from the first and second factor are listed, so that every entry of a Kronecker-structured tensor can be traced back to a unique pair of basis elements without ambiguity.

Lexicographic Ordering Principle

Primary and Secondary Index Roles

The basis ordering treats the index from the first factor as the primary index and the index from the second factor as the secondary index. The composite basis is enumerated by holding the primary index fixed while the secondary index runs through its full range, then advancing the primary index and repeating.

Composite Index Formula

The linear position of a composite basis vector within the ordered list is computed directly from the primary and secondary indices and the extent of the second factor's basis.

Ordering Diagram

Enumeration Pattern

The diagram below shows how basis pairs from a two-element first factor and a three-element second factor are enumerated: the secondary index cycles completely before the primary index advances.

Uniqueness of the Correspondence

Bijection Between Index Pairs and Positions

The basis ordering establishes a one-to-one correspondence between ordered pairs of indices and single positions in the composite enumeration. No two distinct pairs of primary and secondary indices map to the same position, and no position is left without a corresponding pair.

Recovery of Factor Indices From a Position

Given a single composite position, the primary and secondary indices can be recovered uniquely by division and remainder against the extent of the second factor's basis, which confirms that the ordering carries no loss of information relative to the original pair of indices.

Dependence on Factor Order

Non-Commutativity of the Ordering

Swapping the roles of the first and second factor changes which index cycles fastest, so the resulting ordered list of basis pairs is generally different from the original, even though the underlying set of pairs is the same. The ordering is therefore a property of the labeled construction, not merely of the two factor spaces in isolation.

Effect on Matrix Representation

Because coordinates are listed according to this fixed ordering, the matrix representation of a linear map built from a Kronecker product depends on the ordering convention chosen. Adopting a different ordering permutes the rows and columns of the matrix without changing the underlying linear map.

Compatibility With Repeated Kronecker Products

Extension to Three or More Factors

When the basis ordering is extended to a Kronecker product of three or more factors, the same cycling rule applies recursively: the last factor's index cycles fastest, the second-to-last cycles next, and so on, with the first factor's index cycling slowest of all.

Consistency Under Associative Grouping

Whether the factors are grouped from the left or from the right when forming the repeated product, the recursive cycling rule produces the same final ordering of composite basis vectors, so the basis ordering is consistent with the associativity of the Kronecker construction.