4.19.3 Tensor Bilinear Form Symmetry Context

Tensor Bilinear Form Symmetry Context is the set of conditions under which it makes sense to ask whether a bilinear form is symmetric, skew-symmetric, or neither, together with the consequences that follow once that question is meaningfully posed. Symmetry is a property that can only be discussed when a bilinear form's two arguments are drawn from the same space, since comparing f(v, w) to f(w, v) presupposes that w can be substituted into the slot originally occupied by v and vice versa.

When Symmetry Can Be Asked

Requiring a Single Underlying Space

A bilinear form f: V × W → F with V ≠ W has no meaningful notion of symmetry, because swapping the arguments to consider f(w, v) is not even defined unless w ∈ V and v ∈ W, which fails whenever V and W are different spaces, or even different vector spaces of the same dimension without a specified identification between them. Symmetry becomes a sensible question only for forms f: V × V → F on a single space, where both f(v, w) and f(w, v) are simultaneously defined.

Bilinear Pairings Between Distinct Spaces

For a pairing f: V × W → F between genuinely different spaces, such as the evaluation pairing V* × V → F sending (φ, v) ↦ φ(v), the question "is f symmetric" is simply not applicable to the form itself, though one may ask instead whether some derived or associated form on a single space, obtained after identifying V and W, has a symmetry property; the evaluation pairing itself has no symmetric or skew-symmetric classification, only forms V × V → F or W × W → F built from it, if any, would.

The Symmetric Case

Definition and Matrix Signature

Once f: V × V → F is given, f is symmetric if f(v, w) = f(w, v) for all v, w ∈ V, equivalently if its matrix satisfies A = A^T in any basis. Symmetric forms correspond to quadratic forms q(v) = f(v, v) and, over the real numbers, are further classified by signature, the number of positive, negative, and zero eigenvalues of a diagonalized representative.

Diagonalizability

Every symmetric bilinear form on a finite-dimensional space over a field of characteristic not 2 admits a basis in which its matrix is diagonal, obtained by an iterative process resembling Gram–Schmidt orthogonalization adapted to the form rather than to a fixed inner product; this diagonal form makes the form's rank and, over the reals, its signature immediately visible.

The Skew-Symmetric and Alternating Case

Definition and Matrix Signature

f is skew-symmetric if f(v, w) = -f(w, v) for all v, w, equivalently A = -A^T. In characteristic not 2, skew-symmetry is equivalent to f being alternating, meaning f(v, v) = 0 for all v; in characteristic 2 alternating is the strictly stronger and more useful condition, since skew-symmetry there coincides with symmetry.

Normal Form

Every alternating bilinear form on a finite-dimensional space admits a basis in which its matrix is a direct sum of 2 × 2 blocks of the standard form

together with a zero block accounting for the radical, meaning rank alone classifies alternating forms up to congruence, with no analogue of a signature invariant.

Forms That Are Neither Symmetric Nor Skew-Symmetric

Decomposition Into Symmetric and Alternating Parts

Any bilinear form f: V × V → F in characteristic not 2 splits uniquely as f = f_sym + f_alt, where f_sym(v,w) = (f(v,w)+f(w,v))/2 is symmetric and f_alt(v,w) = (f(v,w)-f(w,v))/2 is alternating. A form's symmetry context is therefore not a binary classification but a decomposition, with the "purely symmetric" and "purely skew-symmetric" cases being the two extremes where one of the two parts vanishes identically.

Loss of Symmetry Under Change of the Underlying Pairing

If a bilinear form is built by composing a symmetric form with a linear map that is not self-adjoint, the resulting form generally loses the symmetry of the original: for a symmetric form g on V and a linear map A: V → V, the form f(v,w) = g(Av, w) is symmetric only when A is self-adjoint with respect to g; this illustrates that symmetry is a property of the specific form, sensitive to how it was constructed, not an automatic inheritance from related forms.

Symmetry Context in Relation to the Tensor Product

Symmetric and Alternating Subspaces of V* ⊗ V*

Under the identification of bilinear forms on V with elements of V* ⊗ V*, the symmetric forms correspond exactly to the subspace Sym²(V*) and the alternating forms to ⋀²(V*), with V* ⊗ V* = Sym²(V*) ⊕ ⋀²(V*) in characteristic not 2. The symmetry context of a bilinear form is thus visible directly at the level of the tensor product: it records which summand, or combination of summands, the corresponding tensor occupies.

Why the Distinction Matters Downstream

Whether a bilinear form is treated as symmetric or alternating governs which further algebraic structures it is compatible with: symmetric forms give rise to orthogonal groups and quadratic form theory, while alternating non-degenerate forms give rise to symplectic groups and symplectic geometry; the two theories diverge sharply, so establishing the correct symmetry context for a given bilinear form is a prerequisite before applying either body of theory to it.

Practical Determination of Symmetry Context

Checking via the Matrix

For a bilinear form on a finite-dimensional space, symmetry context is determined by direct inspection of its matrix A in any basis: computing A - A^T and checking whether it vanishes (symmetric), whether A + A^T vanishes (alternating), or neither, gives a complete and basis-independent answer, since congruence transformations A ↦ P^T A P preserve both the vanishing of A - A^T and of A + A^T.

Checking via Defining Formulas

For bilinear forms given by an explicit formula rather than a matrix, such as f(v,w) = ∫ v'(x)w(x)dx on a function space, symmetry context is verified directly from the formula, often via integration by parts or algebraic rearrangement, before any matrix representation is even introduced, confirming that the symmetry question is meaningful and answerable at the level of the form itself, prior to any choice of basis or coordinates.