3.7.4 Tensor Vector Covector Component Contraction

Tensor Vector Covector Component Contraction explores how tensors combine vectors and covectors through component-wise operations in multilinear algebra.

Tensor Vector Covector Component Contraction is the operation of summing the product of a vector's contravariant components against a covector's covariant components over a shared index, and it is the concrete mechanical procedure that realizes the abstract vector-covector pairing once both objects are expressed in coordinates. Written v^i f_i, this contraction is the simplest possible instance of general tensor index contraction, involving exactly one upper index and one lower index, and it serves as the template from which contraction of larger, higher-rank tensors is built by repetition.

The Contraction Procedure

Matching an Upper Index to a Lower Index

Given a vector with contravariant components v^i and a covector with covariant components f_i, both indexed over i = 1, ..., n relative to a chosen basis and its dual basis, contraction pairs the upper index i on v^i with the lower index i on f_i and sums over all values of i:

\sum_{i = 1}^{n} v^{i} f_{i} = v^{i} f_{i}

where the right-hand side uses the Einstein summation convention to suppress the explicit summation sign. The result of this contraction is the scalar v^i f_i, matching the value of the pairing <v, f>.

Why the Index Must Repeat as Upper and Lower

Contraction is only defined, in the sense of producing a basis-independent result, when the repeated index appears once as an upper index and once as a lower index. Summing v^i f^i, with both indices upper, or v_i f_i, with both indices lower, would not correspond to a coordinate-free tensor operation, since the transformation behaviors of the two component arrays would not cancel under a change of basis.

Contraction as Index Removal

Reduction of Free Indices

Before contraction, v^i carries one free upper index and f_i carries one free lower index, for a combined total of two free indices across the two objects. After contraction, both indices are summed away and no longer appear as free indices in the result, leaving an expression with zero free indices, which identifies the result as a scalar.

General Pattern for Higher-Rank Tensors

The same index-removal pattern generalizes directly. Contracting a (p, q) tensor T with a (1, 0) vector v along one particular upper slot, say the first, produces a tensor of type (p - 1, q):

S_{j_{1} \dots j_{q}}^{i^{2} \dots i^{p}} = T_{j_{1} \dots j_{q}}^{k^{1} i^{2} \dots i^{p}} v_{k^{1}}

showing the elementary vector-covector contraction embedded as the local operation performed at one particular pair of index slots, while the remaining indices of T are left untouched.

Distinction from Outer Product

No Contraction: The Outer Product

Contraction should be contrasted with the outer product, which multiplies components without summing any shared index, producing a strictly higher-rank object. The outer product of v^i and f_j is v^i f_j, a (1, 1) tensor with two free indices, i and j, distinct from one another:

W_{j}^{i} = v^{i} f_{j}

Contraction as a Trace of the Outer Product

The full contraction v^i f_i can be recovered from the outer product W^i_j = v^i f_j by setting j = i and summing, which is exactly the trace operation applied to W:

W_{i}^{i} = v^{i} f_{i}

This identity shows that contraction and the trace operation are the same underlying idea, applied here in the smallest possible case of a (1, 1) tensor built from a single vector and a single covector.

Basis Independence of the Contraction Result

Cancellation Under Change of Basis

As with the evaluation operation itself, the contraction result v^i f_i is unchanged under a change of basis, because the transformation matrix acting on v^i and the inverse transformation matrix acting on f_i cancel exactly in the summed product, leaving the scalar invariant. This basis independence is what makes contraction a legitimate tensorial operation rather than an artifact of a particular coordinate choice.

Diagrammatic Summary

The diagram shows the shared index i linking the upper index of v to the lower index of f, being summed away entirely to leave the scalar contraction result.