13.9.4 Tensor Inner Product Scalar Result

The tensor inner product yields a scalar result by combining vector spaces, revealing geometric and algebraic relationships through contraction of indices.

Tensor Inner Product Scalar Result is the single numerical value produced once an inner product contraction between two tensors has been fully carried out, representing the completed output of the operation rather than any of the intermediate index structure, metric input, or component summation used to compute it. It designates the scalar itself, together with the properties it inherits from the bilinear and contraction-based construction that produced it.

Conceptual Basis

The Result as a Completed Pairing

An inner product contraction takes two tensors, applies whatever metric input is necessary to match variances, and sums over every paired index until no free indices remain. The scalar result is precisely this final value, considered independently of the specific tensors, metric, or basis used along the way.

Absence of Residual Index Structure

Because the inner product contraction is constructed so that every index of both participating tensors is consumed, the scalar result carries no directional or component-wise information and cannot, by itself, be used to reconstruct either of the original tensors without additional data.

Distinction From Partially Contracted Intermediate Values

If an inner-product-like contraction is applied to only some of the available index pairs on tensors of higher rank, the intermediate output retains free indices and does not yet constitute the scalar result; the scalar result specifically refers to the case where the contraction is carried through completely.

Formal Description

General Form

For two tensors $u^{i}$ and $v^{j}$ paired via a metric input $g_{i j}$ , the scalar result is:

s = g_{i j} u^{i} v^{j}

with $s$ denoting a single number belonging to the field over which the tensors are defined.

Basis Independence of the Value

If the same two tensors and metric are re-expressed in a different basis, the scalar result computed from the transformed components equals the scalar result computed in the original basis:

g_{i' j'} u^{i'} v^{j'} = g_{i j} u^{i} v^{j}

confirming that the scalar result is one fixed number independent of coordinate choice.

Sign and Magnitude

Depending on the signature of the metric input, the scalar result may be positive, negative, or zero even for nonzero tensors, with a positive definite metric guaranteeing a nonnegative result when a tensor is paired with itself, while an indefinite metric permits results of either sign.

Properties

Bilinearity Inherited From the Construction

The scalar result depends linearly on each of the two tensors supplied to the inner product separately, so scaling either tensor by a constant scales the scalar result by that same constant, and summing tensors before contraction produces a scalar result equal to the sum of the individual results.

Symmetry or Antisymmetry Depending on Context

Whether the scalar result is unchanged or reversed in sign upon exchanging the two tensors depends on the symmetry properties of the metric input used, with a symmetric metric yielding an unchanged result and an antisymmetric metric yielding a sign reversal.

Vanishing as an Orthogonality Criterion

A scalar result equal to zero indicates that the two tensors are orthogonal with respect to the metric input used, providing a direct scalar criterion for a geometric relationship between the tensors that would otherwise require comparing their full component structure.

Practical Significance

Use as a Measurable Quantity

Because the scalar result carries no remaining tensorial structure, it can be directly compared, ranked, or combined with other scalars using ordinary arithmetic, making it the natural quantity used whenever a single measurable number is needed from a pair of tensors.

Foundation for Norms and Distances

Applying the inner product contraction to a tensor paired with itself and taking the scalar result as input to a square root, when the result is nonnegative, produces the standard notion of a tensor's norm, from which distances and angles are subsequently defined.

Verification Through Basis Change

Recomputing the scalar result after an explicit change of basis serves as a practical check on the correctness of a tensor computation, since any error in applying the transformation law to the underlying tensors would typically cause the recomputed scalar to disagree with the original value.