12.10.5 Tensor Substitution Result Type Change
Tensor substitution can alter result types, revealing how algebraic structures transform under different operational contexts.
Tensor Substitution Result Type Change is the specific alteration in a tensor's type, from to either or , that results from performing a single slot substitution, contrasting with the type-preserving behavior of operations such as addition, subtraction, scalar multiplication, and negation.
Describing the Change
Reduction Depending on the Slot Kind
Unlike addition, subtraction, scalar multiplication, and negation, all of which leave a tensor's type entirely unchanged, slot substitution deliberately alters the type by removing exactly one slot. If the filled slot was a vector slot, the covariant count decreases by one, changing the type from to . If instead the filled slot was a covector slot, the contravariant count decreases by one, changing the type to .
A Deliberate Departure from Type Preservation
This type change is not a defect or an inconsistency but an intentional feature of the substitution operation, reflecting the fact that supplying one of the tensor's required arguments genuinely reduces how many further arguments the resulting object still awaits.
Contrast with Type-Preserving Operations
Addition, Subtraction, and Scalar Operations Leave Type Fixed
Operations such as tensor addition, subtraction, scalar multiplication, and negation are specifically defined to require, or automatically produce, tensors of the identical type as their operands, since these operations act uniformly across every component without altering the underlying index structure.
Substitution Specifically Targets the Index Structure
Substitution, by contrast, operates directly on the index structure itself, contracting one particular index of the tensor against a supplied argument and thereby eliminating that index from the resulting object's structure, which is precisely why its result necessarily differs in type from its input.
Consequences of the Type Change
The Result Belongs to a Different Vector Space of Tensors
Because its type differs from that of the original tensor, the result of a substitution belongs to a different vector space of tensors than the one containing the original tensor. It cannot, for instance, be directly added to the original tensor, since addition requires matching type.
Compatibility with Further Type-Matched Operations
Once the type change from substitution has occurred, the resulting tensor can be freely combined, using addition, subtraction, or scalar multiplication, with any other tensor sharing its new, reduced type, following the ordinary rules governing those operations for tensors of that type.
Tracking Type Through Repeated Substitution
Cumulative Effect of Multiple Substitutions
Performing several slot substitutions in succession accumulates the type change correspondingly, with each individual substitution reducing either the contravariant or covariant count by one, depending on the kind of slot addressed, until eventually, after enough substitutions, the type reaches , corresponding to a plain scalar with no remaining slots at all.