9.4.2 Tensor Product Basis Multi Index Address

Tensor Product Basis Multi Index Address is the identification of each basis element of a tensor product space by an ordered tuple of individual indices, one drawn from each factor basis, so that the tuple acts as a unique address locating that basis element within the full family of tensor product basis elements; it is the addressing scheme that turns the abstract combinatorial process of forming all possible tensor products of factor basis vectors into an organized, lookup-able indexing system.

What a Multi Index Address Identifies

One Tuple, One Basis Element

Given factor bases {e_{i}} for one space and {f_{j}} for another, the tensor product basis element e_i ⊗ f_j is addressed by the pair (i, j). No other pair addresses that same basis element, and no basis element is left without an address, so the multi index address establishes a one-to-one correspondence between tuples of factor indices and basis elements of the tensor product space.

Extension to Any Number of Factors

The same addressing scheme extends to a tensor product of any number of factors. A basis element e_{i₁} ⊗ e_{i₂} ⊗ ⋯ ⊗ e_{iₖ} is addressed by the full tuple (i₁, i₂, …, iₖ), with one entry contributed by each factor, and the length of the tuple always equal to the number of factors in the product.

Why an Address Is Needed

The Basis Family Is Too Large to List by a Single Index

A tensor product basis contains one element for every possible combination of factor indices, and the number of such combinations grows as the product of the factor dimensions. Referring to each element by a single flat label would obscure which factor contributed which part of the combination, so the multi index address is used instead, preserving the origin of every component of the tuple.

The Address Encodes the Factorized Structure Directly

Because each entry of the tuple corresponds to exactly one factor, the multi index address makes the factorized origin of a basis element visible at a glance: reading off entry k of the address immediately gives the index used in the k-th factor basis, without requiring any further decoding.

Ordering and Flattening of Multi Index Addresses

Lexicographic Ordering

When the tensor product basis elements must be listed or stored in a definite sequence, multi index addresses are typically ordered lexicographically: addresses are compared entry by entry from the first position onward, with the first differing entry deciding the order. This convention gives a fixed, unambiguous sequence to an otherwise unordered family of tuples.

Collapsing a Multi Index Into a Single Linear Index

A multi index address (i₁, …, iₖ) over factor dimensions n₁, …, nₖ can be collapsed into a single linear address using a mixed-radix formula, allowing the tensor product basis to be stored in an ordinary one-dimensional array while the multi index address remains recoverable from the linear position by the inverse computation.

Diagram of a Multi Index Address

Consequences of Using Multi Index Addresses

Component Arrays Inherit the Same Addressing

Because each tensor product basis element is addressed by a multi index, the coefficients of a tensor expanded in that basis are naturally indexed by the same multi index, so a tensor's component array is itself organized as a multi-dimensional array whose entries are located exactly as the corresponding basis elements are addressed.

Operations Respect the Address Structure

Contraction, symmetrization, and other operations performed on a tensor expressed in a tensor product basis act on specific entries of the multi index address, changing or summing over particular positions of the tuple while leaving the rest untouched; this is possible only because the multi index address keeps the contribution of every factor distinctly and permanently identifiable.