12.10.2 Tensor Substitution Argument Insertion

Tensor Substitution Argument Insertion is the actual act of placing a specific vector or covector value into a tensor's identified target slot, carrying out the contraction implied by that placement and producing the resulting lower-rank tensor as the concrete outcome of the insertion.

The Insertion Process

From Identification to Execution

While identifying the target slot specifies which position within a tensor's argument list will be filled, argument insertion is the subsequent step of actually supplying a concrete vector or covector value into that identified position and carrying out the resulting computation. Insertion is the operational, computational counterpart to the purely positional act of target slot identification.

Formal Expression of Insertion

Given a tensor $$A$$ and a chosen target vector slot at position $$r$$, inserting a specific vector $$v$$ produces:

with summation implied over the repeated index $$k$$, representing the actual numerical contraction carried out by the insertion.

Requirements Before Insertion Can Proceed

Argument Compatibility Must Already Hold

Insertion is only meaningful once the vector or covector being inserted has already been confirmed to belong to the correct underlying space and to be of the correct kind expected by the target slot, since insertion itself performs no additional validation beyond carrying out the contraction.

A Definite Target Slot Must Already Be Identified

Insertion presupposes that a specific target slot has already been unambiguously identified, since inserting an argument without a clearly designated slot would leave the intended contraction undetermined.

Effects of the Insertion

Reduction in Rank

Each act of insertion reduces the rank of the tensor by exactly one, removing the filled slot from the resulting object's list of open positions while leaving every other slot unaffected in kind or dimension.

Preservation of Multilinearity in Remaining Slots

The tensor resulting from insertion remains multilinear with respect to whatever slots are still open, since insertion into one slot does not disturb the linear dependence of the tensor on any of its other arguments.

Repeated Insertion

Sequential Insertions Reducing to a Scalar

Performing insertion repeatedly, once for each remaining open slot in turn, eventually reduces the original tensor entirely to a scalar, with each successive insertion contracting one additional index until none remain.

Insertions Commute Across Different Slots

Because each insertion contracts a distinct index of the original tensor, performing two insertions into two different slots produces the same final result regardless of the order in which those two insertions are carried out.

Illustration