13.6.2 Tensor Full Contraction Scalar Result

Tensor Full Contraction Scalar Result is the outcome of contracting all indices of a tensor, yielding a scalar value through multilinear algebra operations.

Tensor Full Contraction Scalar Result is the single numerical value produced by carrying out the full contraction operation, representing the outcome of summing a tensor's components over an all slot pairing and standing as an invariant quantity independent of any particular choice of basis.

Identity of the Scalar Result

A Value Without Index Structure

The scalar result carries no free indices whatsoever, distinguishing it from the tensors produced by partial contraction, since every index of the original tensor has been consumed by the all slot pairing underlying the full contraction operation.

\sum_{i = 1}^{n} \sum_{j = 1}^{n} T_{i j}^{i j} = c

An Order-Zero Object

The scalar result corresponds precisely to the order-zero outcome described within tensor scalar result scope, sharing that scope's defining property of carrying no remaining index positions once the operation has been completed.

The Defining Invariance Property

Unchanged Under Any Change of Basis

The scalar result remains numerically identical no matter which basis is used to express the components of the original tensor before the full contraction operation is carried out, a property that follows from the tensor transformation law once every index responsible for that law's transformation matrices has been eliminated.

\tilde{c} = c

Contrast with Basis-Dependent Components

Unlike the individual components of the original tensor, which take different numerical values depending on the chosen basis, the scalar result offers a single, basis-independent number that summarizes a specific aggregate property of the original tensor.

Computing the Scalar Result

Summation Over the Full Summation Set

The scalar result is obtained by carrying out the nested summation defined by the summation set associated with the chosen all slot pairing, adding together one term for every combination of values the several summation indices may jointly take.

c = \sum_{i = 1}^{n} \sum_{j = 1}^{n} T_{i j}^{i j}

Dependence on the Chosen All Slot Pairing

Because a tensor may admit more than one distinct all slot pairing, the specific scalar result obtained generally depends on which pairing was selected, so identifying the pairing used remains necessary to interpret which particular scalar result has been computed.

Verification of the Scalar Result

Simplified Invariance Check

Verifying the invariance of a full contraction scalar result requires only confirming exact numerical agreement across two different bases, a comparatively simple check relative to the transformation-law comparison required for results retaining free indices.

Confirmation of Complete Slot Coverage

Verification of a claimed scalar result additionally confirms that the slot pair set used genuinely constitutes an all slot pairing, since an incomplete pairing would leave residual free indices and would not correspond to a genuine scalar result despite superficially resembling one.

Relationship to Tensor Operation Notation

The scalar result is denoted in tensor operation notation by an expression stripped of every index symbol once all matched upper-lower pairs have been summed away, so that the complete absence of any indexed symbol in the final notated form signals that the expression represents a full contraction scalar result.