14.15.2 Tensor Map Product Composition Ordering

Tensor Map Product Composition Ordering defines how tensor maps combine, structuring operations to maintain algebraic consistency in multilinear algebra.

Tensor Map Product Composition Ordering is the discipline of tracking the exact left-to-right order in which factor maps are composed within each individual factor when applying the composition rule for tensor products of maps, since composition of operators generally does not commute and reversing the order within a factor can change the resulting combined operator.

Why Ordering Matters

Composition Is Not Commutative

For a single factor space, composing one operator after another generally produces a different result from composing them in the reverse order, so the composition rule for tensor products of maps must preserve whichever order was originally specified for each factor.

S_{1} \circ T_{1} \neq T_{1} \circ S_{1}

Consequence for the Composition Rule

Because the composition rule states that composing two combined operators equals the tensor product of the composed factor maps, the order in which S₁ is composed with T₁, and the order in which S₂ is composed with T₂, must both match the order in which the original combined operators were composed.

(S_{1} \otimes S_{2}) \circ (T_{1} \otimes T_{2}) = (S_{1} \circ T_{1}) \otimes (S_{2} \circ T_{2})

Diagram Contrasting Correct and Incorrect Ordering

Preserving Versus Reversing the Order Within a Factor

The diagram below contrasts the correctly ordered composition within each factor with an incorrectly reversed ordering, which generally produces a different combined operator.

Ordering Across Different Factors

Independence Between Separate Factor Slots

While the order within a single factor must be preserved carefully, the order in which computations for different factor slots are carried out relative to each other does not matter, since the factor operators act on entirely separate components and do not interfere with one another.

No Cross-Factor Ordering Constraint

There is no meaningful sense in which the operator on the first factor must be applied before or after the operator on the second factor, since the two operators act simultaneously on different components of an elementary tensor rather than sequentially on the same component.

Ordering in Chains of Several Composed Operators

Preserving a Longer Chain's Order

When several combined operators are composed in a chain, the composition ordering rule extends by preserving the original left-to-right order of composition within each individual factor's corresponding chain, even as the overall chain of combined operators grows longer.

(R_{1} \otimes R_{2}) \circ (S_{1} \otimes S_{2}) \circ (T_{1} \otimes T_{2}) = (R_{1} \circ S_{1} \circ T_{1}) \otimes (R_{2} \circ S_{2} \circ T_{2})

Verifying an Extracted Chain Against the Original

Given a long chain of combined operators, the factor-level chain extracted for a single factor can be checked against the original chain simply by confirming that the two chains list the same maps in the same order, ignoring the other factor's contributions entirely.

Special Cases Where Ordering Is Immaterial

Commuting Factor Operators

If the individual operators within a single factor happen to commute with one another, the ordering within that factor slot becomes immaterial for that specific factor, though this commuting property must be verified separately and cannot be assumed in general.

Identity Operators Anywhere in the Chain

Regardless of ordering concerns, an identity operator appearing anywhere within a chain for a given factor can always be removed from that chain without affecting the result, since composing with the identity, at any position in the ordering, leaves the rest of the chain unchanged.

Matrix-Level Ordering

Order of Matrix Multiplication Within Each Factor

At the level of matrices, composition ordering corresponds to the requirement that the order of matrix multiplication within each factor's chain of matrices must match the order of composition of the corresponding combined operators, since matrix multiplication is likewise not commutative in general.

Consequence for Numerical Computation

A numerical implementation that composes combined operators by multiplying factor matrices must therefore multiply the matrices for each factor in the correct order, since silently reversing the multiplication order within a factor would generally produce an incorrect composite matrix.