14.13.1 Tensor Map Product Triple Factor Grouping

Tensor Map Product Triple Factor Grouping is a structured approach to combining tensor maps through triple factor interactions in algebraic frameworks.

Tensor Map Product Triple Factor Grouping is the specific case of associativity concerned with exactly three linear maps entering a tensor product, examining the two distinct ways of inserting a single pair of parentheses among three factors and confirming that both groupings yield the same combined operator.

The Two Possible Groupings

Left Grouping

The left grouping combines the first two maps into a single combined operator first, and then combines that result with the third map.

(T_{1} \otimes T_{2}) \otimes T_{3}

Right Grouping

The right grouping instead combines the last two maps into a single combined operator first, and then combines the first map with that result.

T_{1} \otimes (T_{2} \otimes T_{3})

Verifying the Two Groupings Agree

Action on a Triple Elementary Tensor

Both groupings, when applied to an elementary tensor built from one vector per factor space, apply each individual map to its own corresponding vector and recombine the three resulting images into a single elementary tensor, producing an identical outcome under either grouping.

[(T_{1} \otimes T_{2}) \otimes T_{3}] (v_{1} \otimes v_{2} \otimes v_{3}) = T_{1} (v_{1}) \otimes T_{2} (v_{2}) \otimes T_{3} (v_{3})

Extension From Elementary to General Tensors

Because both groupings agree on every elementary tensor and both extend linearly to the full tensor product space, the two groupings must agree on every general tensor as well, since a linear map is completely determined by its values on a spanning set.

Diagram of the Two Groupings

Left and Right Groupings Converging

The diagram below shows the left grouping and the right grouping of three maps converging to the same final combined operator.

Intermediate Space Identification

Distinct Intermediate Tensor Spaces

The left grouping first forms a tensor product of the first two factor spaces, while the right grouping first forms a tensor product of the last two factor spaces, so the two groupings pass through different intermediate spaces before reaching the same final triple tensor product space.

Natural Identification Between Groupings

The two possible ways of parenthesizing the tensor product of three spaces are connected by a natural identification that matches elementary tensors formed one way to the corresponding elementary tensors formed the other way, and it is this identification that allows the two groupings of maps to be regarded as literally equal rather than merely similar.

Matrix-Level Verification

Kronecker Product Associativity for Three Matrices

Once bases are fixed, the triple factor grouping translates directly into the associativity of the Kronecker product for three matrices, where grouping the first two factor matrices before combining with the third produces the same composite matrix as grouping the last two factor matrices before combining with the first.

(A \otimes B) \otimes C = A \otimes (B \otimes C)

Practical Freedom in Computation Order

Because both groupings are guaranteed to agree, a computation involving three factor matrices can compute either the left grouping or the right grouping first, choosing whichever intermediate matrix is smaller or more convenient, without affecting the final composite matrix.

Role as the Base Case for Longer Products

Foundation for Groupings of Four or More Factors

The agreement between the two groupings of three factors serves as the foundational case used to establish that every possible grouping of four or more factors also agrees, since any longer grouping can be reduced to a sequence of triple groupings applied repeatedly.

Building Larger Groupings From the Triple Case

A grouping of four factors, for instance, can be decomposed into two applications of the triple factor grouping result, first combining three of the four factors under either grouping and then combining the outcome with the remaining fourth factor.