12.10.3 Tensor Substitution Remaining Slot Structure
Tensor Substitution Remaining Slot Structure refers to unresolved indices after substitution, key for tensor transformations and contractions in algebra.
Tensor Substitution Remaining Slot Structure is the description of the type, rank, and arrangement of argument positions that persist in a tensor after one of its slots has been filled through substitution, characterizing exactly what kind of multilinear object is left over once a single argument has been inserted.
Determining the Remaining Structure
Loss of Exactly One Slot
When a substitution fills one slot of a tensor of type , the remaining slot structure has exactly one fewer position of the corresponding kind. Filling a vector slot reduces the covariant count by one, yielding a remaining structure of type , while filling a covector slot instead reduces the contravariant count by one, yielding type .
All Other Slots Remain Untouched
Every slot other than the one targeted by the substitution retains exactly its original kind, whether vector-accepting or covector-accepting, and its original dimension, unaffected by the substitution performed elsewhere. Only the count of slots of the relevant kind decreases, and only by exactly one.
Renumbering Convention for Remaining Slots
Reindexing After a Slot Is Removed
Once a target slot is filled, the remaining open slots are typically renumbered to reflect their new positions within the reduced structure, since the original numbering included the slot that has now been eliminated. A slot that was originally third among the vector slots, for instance, becomes second if the first vector slot was the one filled by substitution.
Preserving Relative Order
Despite this renumbering, the relative order of the remaining slots to one another is preserved: any two slots that were in a particular order before substitution remain in that same relative order afterward, only shifted in absolute numbering to account for the removed slot.
Multilinear Nature of the Remaining Structure
Genuine Multilinearity in the Surviving Slots
The tensor produced by substitution remains multilinear with respect to every one of its remaining slots, since fixing one argument of a multilinear map while leaving the others open produces an object that is still linear in each of those surviving arguments considered individually.
Behaving as an Ordinary Tensor of Reduced Rank
Because the remaining slot structure is itself a well-defined type , the resulting object after substitution can be treated in every respect as an ordinary tensor of that reduced type, fully eligible for further operations such as addition, scalar multiplication, or additional substitution.
Consequences for Further Composition
Compatibility with Subsequent Operations
Because the remaining slot structure is a definite, well-characterized tensor type, it can be combined with other tensors of that same reduced type using addition or subtraction, or scaled by a further scalar multiplication, exactly as any tensor of that type would be.
Guiding Repeated Substitution
Understanding the remaining slot structure after one substitution makes it possible to correctly identify target slots for any further substitutions, since the numbering and kind of the available slots have shifted according to the rules just described.