15.5.1 Tensor Symmetric Bilinear Argument Pair

Tensor Symmetric Bilinear Argument Pair is the ordered pair of vector inputs supplied to a symmetric bilinear form, considered specifically with respect to the property that exchanging the two members of the pair leaves the value returned by the form unchanged. Although the pair is written in a definite order, u followed by v, the symmetric bilinear form structure guarantees that the pair (u, v) and the pair (v, u) are functionally interchangeable, so the ordering of the pair carries no information about the resulting scalar even though the pair itself is formally an ordered object.

The argument pair is the basic unit on which a symmetric bilinear form acts, and understanding its behavior under exchange is what distinguishes a symmetric form from a general bilinear form, where the order of the two arguments can affect the returned value. Working directly with the argument pair, rather than only with the resulting scalar, makes explicit which manipulations of the inputs are permitted without altering the output.

Order Invariance of the Pair

Statement of Invariance

For a symmetric bilinear form B built from a symmetric tensor T, the argument pair (u, v) satisfies:

for every choice of vectors u and v drawn from the vector space on which B is defined, with no restriction on u and v beyond membership in that space.

Origin of the Invariance in the Component Constraint

Expanding both sides in components, B(u, v) sums T_{ij} u^i v^j while B(v, u) sums T_{ij} v^i u^j, and relabeling the summation indices in the second sum shows it equals the sum of T_{ji} u^i v^j; the two expressions coincide term by term exactly when T_{ij} equals T_{ji}, which is the symmetric component equality constraint applied directly to the argument pair.

The Pair as an Unordered Object in Practice

Equivalence Class Under Exchange

Although (u, v) and (v, u) are distinct as ordered pairs, they belong to the same equivalence class with respect to the value returned by any symmetric bilinear form, since the exchange operation is an involution whose two outcomes are guaranteed identical. This allows the argument pair to be treated as an unordered pair, denoted {u, v}, whenever only the value of the form is of interest.

Diagonal Pairs

A special case of the argument pair occurs when u and v coincide, giving the pair (v, v). Exchange invariance is trivially satisfied in this case, and evaluating the form on such a pair produces the associated quadratic form Q(v) = B(v, v), which depends on a single vector rather than on an argument pair.

Contrast With Non-Symmetric Argument Pairs

Behavior Under a General Bilinear Form

For a bilinear form that is not symmetric, the argument pair (u, v) and its exchange (v, u) generally return different values, since the underlying tensor need not satisfy T_{ij} = T_{ji}. In that setting, the order of the pair is essential information, and the pair cannot be treated as unordered without changing the result.

The Symmetric Case as a Special Constraint

The order invariance of the argument pair under a symmetric bilinear form is therefore not a generic property of bilinear forms but a specific consequence of imposing the symmetric component equality constraint on the underlying tensor; removing that constraint removes the guarantee that exchanging the pair leaves the value unchanged.

Use in Constructing and Verifying Symmetry

Testing Symmetry via Sample Pairs

Because the exchange identity B(u, v) = B(v, u) must hold for every argument pair, testing it on a spanning set of basis vector pairs, such as (e_i, e_j) for all i and j, is sufficient to verify that a given bilinear form is symmetric, since bilinearity extends the identity from basis pairs to arbitrary argument pairs by linearity in each slot.

Building Symmetric Forms From Arbitrary Pairs

Given an arbitrary, not necessarily symmetric bilinear form B, a symmetric form can be constructed by averaging over the argument pair and its exchange, B_sym(u, v) = (B(u, v) + B(v, u)) / 2, which by construction satisfies the order invariance required of a symmetric bilinear argument pair regardless of whether the original form did.