14.17.3 Tensor Map Product Multi Index Arrangement

Tensor Map Product Multi Index Arrangement uses multi-index notation to systematically represent tensor products and multilinear mappings in algebra.

Tensor Map Product Multi Index Arrangement is the scheme for organizing the entries of a combined operator built from three or more factor maps using a full tuple of individual factor indices, one per factor, in place of a single pair of composite row and column numbers, exposing the natural multi-dimensional structure that the flattened composite matrix otherwise conceals.

From a Single Composite Index to a Full Tuple

Motivation for Moving Beyond a Single Pair of Indices

While a composite matrix entry can always be located using a single composite row number and a single composite column number, this pair of numbers hides the fact that the entry is really determined by one row index and one column index from every individual factor; the multi-index arrangement makes that underlying structure explicit.

(i_{1}, i_{2}, \dots, i_{n})

The Full Row and Column Tuples

A row position in the multi-index arrangement is a tuple listing one row index for each of the n factor maps, and similarly a column position is a tuple listing one column index for each of the n factor maps, together specifying a single entry of the combined operator's matrix.

Recovering the Flattened Composite Index

Converting a Tuple Into a Single Number

Given the full row tuple, the single composite row number used in the ordinary flattened matrix is recovered by combining the individual factor indices using the extents of all the later factors as step sizes, generalizing the two-factor decomposition rule.

i = \sum_{k = 1}^{n} (i_{k} - 1) \prod_{l = k + 1}^{n} m_{l} + 1

Recovering the Tuple From the Flattened Index

Conversely, given the single flattened composite row number, the individual factor indices making up the tuple are recovered by repeated integer division and remainder, isolating one factor's index at each step, exactly matching the recursive component index pattern used for many factors.

Diagram of the Multi-Index Arrangement

A Tuple of Factor Indices Identifying a Single Entry

The diagram below shows a tuple of three factor row indices and a tuple of three factor column indices together identifying a single entry of a combined operator built from three factor maps.

Component Formula in Multi-Index Form

Entry as a Product Over All Factors

Using the multi-index arrangement, the component formula for a many-factor combined operator is written as a single product over all factors, with each factor contributing exactly one matrix entry selected by its own pair of tuple indices.

C_{i j} = \prod_{k = 1}^{n} a_{k} (i_{k}, j_{k})

Clarity Gained From the Multi-Index Form

Writing the component formula this way makes immediately visible that the composite entry depends symmetrically on all n factors, each contributing exactly one term to the product, a symmetry that is present but much less visible in the recursively nested two-index decomposition.

Practical Uses of the Multi-Index Arrangement

Direct Indexing in Multi-Dimensional Array Storage

Many computational tools store a many-factor combined operator's data as a multi-dimensional array indexed directly by the full tuple of factor indices, rather than as a flattened two-dimensional matrix, making the multi-index arrangement the natural way to access such storage.

Simplifying Reasoning About Individual Factors

Because the multi-index arrangement isolates each factor's index explicitly, reasoning about how a change to a single factor's map affects the combined operator becomes more direct, since only the term in the product corresponding to that one factor needs to be examined.

Relationship to the Two-Factor Case

Two-Factor Arrangement as a Special Case

The multi-index arrangement for two factors reduces exactly to the ordinary pair of outer and inner indices already used throughout the discussion of the component formula, confirming that the multi-factor generalization is a direct extension rather than an unrelated construction.

Consistency Across Different Numbers of Factors

Every property established for the multi-index arrangement, including the recovery of the flattened index and the component formula written as a product, specializes correctly to the two-factor, three-factor, or any other specific number of factors, without requiring separate treatment for each case.

Symmetry and Reordering Considerations

Reordering the Tuple Positions

Because the multi-index arrangement lists factor indices in a fixed order matching the order of the factors in the original tensor product, reordering the factors themselves corresponds to reordering the positions within every tuple consistently, which must be tracked carefully to keep the arrangement matched to the correct factor maps.

Symmetric Role of Every Factor in the Product Formula

Despite this fixed ordering convention, the multiplicative component formula treats every factor's contribution symmetrically, in the sense that the product of all the individual factor entries does not depend on the order in which the multiplication is carried out, only on which specific entries are selected from each factor.