12.10 Tensor Slot Substitution Operation

Tensor Slot Substitution Operation is the operation of inserting a specific vector or covector into one designated argument slot of a tensor regarded as a multilinear map, thereby producing a new object that no longer requires an input at that slot while remaining ready to accept whatever inputs the tensor's other slots still demand.

Formal Definition

Substituting Into One Slot

For a tensor $$A$$ of type $$(p,q)$$, slot substitution designates a particular slot, say the $$r$$-th vector slot, and replaces it with a specific vector $$v$$:

leaving every other slot open, represented here by the placeholder symbol.

Result as a New Multilinear Object

The substitution produces a new tensor of type $$(p,q-1)$$, since one covariant slot has been filled and no longer needs to be supplied, while the remaining $$p$$ covector slots and $$q-1$$ vector slots remain available to accept further arguments.

Componentwise Realization

Contracting the Chosen Index

In component form, slot substitution corresponds to contracting the tensor's array against the components of $$v$$ specifically along the index associated with the chosen slot:

with the index at position $$r$$ summed against $$v^k$$, and every other index left free in the resulting array $$B$$.

Distinguishing Substitution from Full Evaluation

A Single Slot Filled Rather Than All Slots

Slot substitution specifically concerns filling one particular slot at a time, leaving the operation's result still dependent on whatever other arguments remain. This differs from full tensor evaluation, where every slot is filled simultaneously to produce a single scalar.

Building Toward Full Evaluation Through Repeated Substitution

Performing slot substitution repeatedly, once for each remaining open slot, eventually reduces a tensor all the way down to a scalar, so full evaluation can be understood as the outcome of applying slot substitution one slot at a time until no open slots remain.

Properties of Slot Substitution

Linearity in the Substituted Argument

Because the underlying evaluation operation is multilinear, slot substitution is linear in the vector or covector being substituted, so substituting a linear combination of vectors into a slot yields the corresponding linear combination of the resulting tensors.

Order Independence Among Different Slots

Substituting into two different slots can be performed in either order without changing the final result, since each substitution contracts a distinct index of the original tensor, and contraction along separate indices does not interfere between the two operations.

Illustration