14.17 Tensor Map Product Component Formula

Tensor Map Product Component Formula is the general closed-form expression that computes any single entry of the matrix representing a combined operator directly from one entry of each factor map's matrix, unifying the index decomposition and the multiplicative combination of factor entries into a single explicit rule.

The Complete Formula

Combining Index Decomposition and Multiplication

The component formula states that the composite matrix entry at row i and column j equals the product of the corresponding first factor entry and second factor entry, where the factor-level indices are obtained by decomposing i and j according to the fixed pattern determined by the second factor's extents.

Two Components Working Together

This single formula packages together two ideas developed separately elsewhere: the component index pattern, which specifies how to obtain i₁, i₂, j₁, and j₂ from i and j, and factor component multiplication, which specifies how the two selected entries combine to produce the composite entry.

Reading the Formula in Practice

Locating an Entry From a Composite Position

Given a specific composite row and column, the formula is applied by first decomposing the composite indices into their outer and inner parts, then looking up the corresponding entries in the two factor matrices, and finally multiplying those two entries together.

Example Illustrating the Full Process

For a composite position with row three and column two, decomposition against a second factor of size two by two identifies the outer indices as two and one and the inner indices as one and two, so the composite entry equals the product of the second factor's row-two, column-one entry of the first matrix and the first factor's row-one, column-two entry of the second matrix, taken in the appropriate order.

Diagram of the Component Formula in Action

From Composite Position to a Single Numerical Value

The diagram below traces the full path from a composite position through index decomposition and entry lookup to the final multiplied value.

Relationship to Direct Kronecker Product Construction

Equivalence With Constructing the Full Matrix

Applying the component formula to every valid composite position and collecting the results produces exactly the same matrix as directly constructing the Kronecker product of the two factor matrices, confirming that the formula is a faithful, entry-by-entry description of the same underlying construction.

Advantage of the Formula Over Full Construction

Unlike full construction, the component formula allows any individual entry to be computed on its own, without requiring every other entry of the composite matrix to be computed at the same time, which is especially valuable when only a small number of entries are actually needed.

Behavior at Special Positions

Diagonal Entries

The component formula applied to a diagonal composite position, where the row index equals the column index, requires both the outer indices and the inner indices to coincide, so diagonal composite entries are products of diagonal entries from the two factor matrices.

Entries Involving Identity or Zero Factors

If the first factor map is the identity, its matrix entries are one on the diagonal and zero elsewhere, so the component formula reduces to selecting entries of the second factor matrix directly, scaled by either one or zero depending on whether the outer indices coincide.

Role Within the Broader Framework

Foundation for Sparse and Large-Scale Computation

The component formula underlies practical strategies for working with very large tensor product matrices, since it allows selective computation of only the entries that are actually required, avoiding the memory and time cost of constructing the full composite matrix whenever that is unnecessary.

Connection to Evaluation of Combined Operators

The same formula that computes matrix entries also underlies the coordinate-based evaluation of combined operators on input tensors, since multiplying a coordinate column by the composite matrix requires accessing exactly the entries that the component formula describes.

Extension to Several Factors

Recursive Component Formula for Many Factors

When the combined operator involves three or more factor maps, the component formula extends by recursive index decomposition, isolating one factor's indices at each stage, and the resulting composite entry equals the product of one matrix entry taken from every individual factor map.

Preserving All Established Properties

Every property already established for the two-factor component formula, including its equivalence to full construction, its behavior at diagonal positions, and its usefulness for sparse and selective computation, extends directly to the multi-factor case without requiring any fundamentally new argument.