14.17.1 Tensor Map Product Component Index Pattern

Tensor Map Product Component Index Pattern is the systematic scheme by which a single composite row index and a single composite column index of a combined operator's matrix are decomposed into a pair of outer indices and a pair of inner indices, one pair drawn from each factor, following a fixed, predictable pattern that repeats regularly across the entire composite matrix.

The Decomposition Pattern

Row Index Decomposition

A composite row index i decomposes into an outer row index i₁, associated with the first factor, and an inner row index i₂, associated with the second factor, using the row extent of the second factor's matrix as the step size.

Column Index Decomposition

A composite column index j decomposes in exactly the same manner as the row index, but using the column extent of the second factor's matrix as the step size, producing an outer column index j₁ and an inner column index j₂.

The Repeating Structure of the Pattern

Cycling of the Inner Index

As the composite index increases by one, the inner index cycles through its full range before resetting, while the outer index remains fixed; only once the inner index has cycled completely does the outer index advance by one.

Regularity Across the Whole Composite Matrix

Because this cycling behavior is entirely determined by the fixed extents of the second factor's matrix, the same repeating pattern applies uniformly across every row and every column of the composite matrix, with no irregularities or exceptions.

Diagram of the Repeating Pattern

Inner Index Cycling Beneath a Fixed Outer Index

The diagram below shows the inner index cycling through its full range for a fixed outer index, before the outer index advances and the inner index resets.

Pattern for the Row and Column Indices Considered Together

Independent Decomposition of Rows and Columns

The row index and column index of a composite entry are decomposed completely independently of each other, since the row decomposition uses only the row extent of the second factor while the column decomposition uses only the column extent of the second factor.

Full Specification of a Composite Entry's Position

Combining the outer and inner row indices with the outer and inner column indices fully specifies which single entry of the first factor matrix and which single entry of the second factor matrix contribute, through factor component multiplication, to the value found at that composite position.

Recognizing the Pattern Visually

Block Boundaries Aligning With Outer Index Changes

Because the inner index completes a full cycle before the outer index advances, the points at which the outer index changes correspond exactly to the boundaries between the blocks of the composite matrix, tying the index pattern directly to the block structure discussed elsewhere.

Diagonal Entries Within the Pattern

A composite entry lies on the main diagonal precisely when its row index equals its column index, which under the index pattern requires both the outer indices to coincide and the inner indices to coincide simultaneously.

Using the Pattern for Efficient Indexing

Direct Computation Without Iteration

Because the pattern is given by simple integer division and remainder operations, the outer and inner indices corresponding to any composite position can be computed directly, without needing to iterate through the composite matrix in order to locate them.

Reverse Lookup From Factor Indices to Composite Position

The same pattern can be run in reverse: given a desired outer index and inner index for a factor, the corresponding composite index is recovered directly by the formula relating composite indices to their decomposed parts, allowing composite positions to be located from factor-level information.

Extension to Several Factors

Recursive Index Pattern With Many Factors

When the combined operator involves three or more factor maps, the component index pattern is applied recursively: the composite index is first decomposed against the combined extent of all remaining factors, and the resulting inner index is then decomposed again against the extent of the next factor, continuing until every individual factor's index has been isolated.

Consistency of the Recursive Pattern

At every stage of this recursive decomposition, the same cycling behavior observed in the two-factor case applies locally between the current factor being isolated and the combined extent of all factors that remain, ensuring the pattern generalizes consistently regardless of how many factors are involved.