13.3.5 Tensor Contraction Result Tensor

Tensor contraction results in a new tensor by summing over paired indices, reducing rank and altering component structure through index pairing and summation.

Tensor Contraction Result Tensor is the tensor produced as output once a contraction operation has been carried out, characterized by the free indices inherited from the input tensor together with the numerical values obtained by summing the input's components over the selected pair of contracted indices.

Structural Composition of the Result

Indices Inherited from the Input

The result tensor retains every index of the input tensor other than the two consumed by the contraction, with each retained index preserving the variance and dimension it possessed on the input, since contraction does not alter the character of indices it does not act upon.

T_{i k}^{i l} \to S_{k}^{l}

Order Reduced by Exactly Two

The result tensor possesses an order exactly two less than the input tensor, reflecting the removal of one contravariant and one covariant index accomplished by the single index contraction operation.

order (S) = order (T) - 2

Numerical Values of the Result

Determined by Index Summation

Each component of the result tensor is obtained by summing the corresponding components of the input tensor over every value of the contracted summation index, with the free indices held fixed at the values identifying that particular component of the result.

S_{k}^{l} = \sum_{i = 1}^{n} T_{i k}^{i l}

Dependence on the Basis of Expression

The numerical values of the result tensor are expressed relative to the same basis used to express the input tensor, so that changing the basis of the input before contraction and changing the basis of the result after contraction produce consistent, corresponding descriptions of the same underlying result.

Transformation Behavior of the Result

Inheriting a Valid Transformation Law

The result tensor transforms under a change of basis according to the standard tensor transformation law associated with its remaining free indices, a property guaranteed by the fact that contraction is constructed specifically to preserve the transformation behavior implied by the input tensor's own indices.

{\tilde{S}}_{k}^{l} = \frac{\partial x^{m}}{\partial {\tilde{x}}^{k}} \frac{\partial {\tilde{x}}^{l}}{\partial x^{p}} S_{m}^{p}

The Special Case of a Scalar Result

When the input tensor's order equals two, so that no free indices remain, the result tensor reduces to a scalar, carrying no index structure and remaining entirely unchanged under any change of basis.

Subject to Output Verification

Structural Confirmation

The result tensor is the object examined during the output verification stage of the general verification procedure, with checks confirming that its order and index arrangement match what is expected given the verified input and the selected slot pair.

Confirmation of Invariance

Invariance verification applied to the result tensor confirms that it transforms correctly under a change of basis, or, in the case of a scalar result, that its numerical value remains identical regardless of the basis used to compute it.

Relationship to Tensor Operation Notation

The result tensor is denoted in tensor operation notation by an expression retaining only the free index symbols of the original tensor, with the previously repeated summation index symbol no longer appearing, so that the absence of that symbol from the final notated form directly signals the reduction accomplished by the contraction.