3.8.4 Tensor Natural Pairing Coordinate Expression

Tensor Natural Pairing Coordinate Expression is the general formula for computing the natural pairing f(v) once both the vector and the covector are expressed in coordinates relative to a chosen basis of V and a chosen basis of V*, covering both the special case where the two bases are dual to one another and the more general case where they are not. While the dual-basis case reduces to the simple sum f_i v^i, the coordinate expression in full generality involves an explicit pairing matrix that records how the chosen basis of V* actually pairs with the chosen basis of V.

The General Pairing Matrix

Setting Up Independent Bases

Let e_1, ..., e_n be a basis of V, and let g^1, ..., g^n be some basis of V*, not necessarily the dual basis of e_1, ..., e_n. Define the pairing matrix P by its entries

recording the scalar obtained by pairing the a-th basis covector with the i-th basis vector. Unlike the dual-basis case, this matrix is, in general, not the identity matrix.

The Coordinate Formula in Full Generality

Writing f = f_a g^a in the chosen covector basis and v = v^i e_i in the chosen vector basis, bilinearity gives

which reduces to the familiar f_i v^i precisely when P is the identity matrix, that is, precisely when g^a = e^a is the dual basis of e_1, ..., e_n.

Recovering the Dual-Basis Special Case

When the Pairing Matrix Is the Identity

If the covector basis is deliberately chosen to be dual to the vector basis, P^a_i = δ^a_i, and the coordinate expression collapses to the single-index sum described by the canonical form of the pairing. This is why, in most practical work, the dual basis is used by default: it is the choice of covector basis that makes the coordinate expression as simple as possible.

General Bases Still Give a Valid, Basis-Independent Scalar

Even when P is not the identity, the value of f_a v^i P^a_i computed from the general formula is still exactly equal to the coordinate-free pairing f(v), and it still does not depend on any further choice of basis for expressing the change-of-basis relationships; only the specific numerical value of P changes depending on which pair of bases was selected.

Worked Example with a Non-Dual Basis

Setting Up a Concrete Case

Let V = R^2 with standard basis e_1 = (1, 0), e_2 = (0, 1), so the dual basis is e^1(x, y) = x, e^2(x, y) = y. Suppose instead a covector basis g^1 = e^1 + e^2 and g^2 = e^2 is chosen, which is not dual to e_1, e_2.

Computing the Pairing Matrix

so P is the matrix [[1, 1], [0, 1]], not the identity. For f = g^1 (so f_1 = 1, f_2 = 0 in this basis) and v = e_1 (so v^1 = 1, v^2 = 0), the general formula gives f(v) = f_a v^i P^a_i = 1 \cdot 1 \cdot 1 = 1, which agrees with directly computing g^1(e_1) = (e^1 + e^2)(1, 0) = 1.

Practical Guidance for Choosing Coordinates

Preferring Dual Bases for Simplicity

Because using a dual basis eliminates the pairing matrix entirely, it is almost always preferable when performing explicit calculations, and most tensor calculus texts implicitly assume dual bases are in use whenever the simple formula f_i v^i appears without further comment.

When Non-Dual Bases Arise Naturally

Non-dual bases can still arise naturally, for example when a covector basis is fixed by some independent physical or geometric consideration unrelated to a particular basis of V. In such cases, the general coordinate expression with an explicit pairing matrix P remains necessary and correct.

Diagrammatic Summary

The diagram shows the general pairing matrix P mediating between an arbitrary covector basis and an arbitrary vector basis, collapsing to the identity matrix only in the dual-basis special case.