13.3.1 Tensor Contraction Input Tensor

Tensor Contraction Input Tensor is the tensor, or product of tensors, that the contraction operation acts upon, identified as the specific object supplying the index slots from which a contravariant and covariant pair are selected and summed to produce the operation's result.

Requirements the Input Must Satisfy

Presence of at Least One Contravariant and One Covariant Index

The input tensor must possess at least one contravariant index and at least one covariant index, since the contraction operation cannot be defined on a tensor lacking one of these two required index types.

A tensor of this purely contravariant form, for instance, does not qualify as a valid input on its own, since it lacks any covariant index available for pairing.

Sufficient Order for the Intended Selection

The input tensor must carry a sufficient number of index slots to accommodate whatever pairing is intended, so that a proposed contraction referencing a slot beyond the tensor's actual order cannot be applied to that tensor as an input.

The Input as a Single Tensor Versus a Product

A Single Tensor with Both Index Types

In the simplest case, the input to a contraction is a single tensor already possessing both a contravariant and a covariant index among its own slots, allowing the operation to act directly on that one object.

A Product of Separate Tensors Combined First

Alternatively, the input to a contraction may be formed by first taking the tensor product of two or more separate tensors, after which the contraction pairs an index contributed by one factor with an index contributed by another, treating the combined product as the single object the operation acts upon.

Properties of the Input Preserved Through the Operation

Underlying Vector Space

The vector space, and its dual, from which the input tensor's indices are drawn remains the same space associated with every index of the resulting output tensor, since contraction does not alter the underlying space but only removes a pair of indices ranging over it.

Basis of Expression

The basis in which the input tensor's components are expressed carries through to the components of the output tensor, so that a contraction performed on components given relative to a particular basis produces output components relative to that same basis.

Distinguishing the Input from the Output

Higher Order Than the Result

The input tensor always possesses an order exactly two greater than the tensor produced as output, reflecting the fixed reduction associated with a single contraction, a relationship that distinguishes the role of the input from the role of the resulting tensor within the operation.

Subject to Input Verification Before Computation

As the object supplied to the operation, the input tensor is precisely what is examined during the input verification stage of the general verification procedure, with checks such as slot existence, opposite variance, and dimension agreement applied directly to the index positions the input makes available.

Relationship to Tensor Operation Notation

The input tensor is represented in tensor operation notation by the indexed symbol, or product of indexed symbols, appearing before any summation has been carried out, with the specific upper and lower indices displayed on that symbol identifying exactly which slots are available to be selected as the pair for contraction.