14.6.5 Tensor Bilinear Form Product Form Type Relation

Explore how tensor bilinear forms relate product forms through type relations in algebraic structures.

Tensor Bilinear Form Product Form Type Relation is the rule determining the symmetry classification, whether symmetric, alternating, or neither, of a combined bilinear form built by tensoring two bilinear forms, as a function of the symmetry classifications of the two original forms.

The Three Basic Classifications

Symmetric, Alternating, and General Forms

A bilinear form b on V times W, with V equal to W, is called symmetric when b(v,w) equals b(w,v) for all v and w, and alternating when b(v,v) equals zero for every v; a form satisfying neither condition is classified simply as a general bilinear form. The form type relation addresses how these classifications behave under the tensor product combination of two forms.

Restriction to the Case of Matching Argument Spaces

Since symmetry and alternation are properties defined for forms whose two argument spaces coincide, the form type relation applies specifically to the combined form on (V tensor U) times (V tensor U), obtained when the argument pairing identifies both copies of the first space with V and both copies of the second space also with V, rather than to the general case of four distinct spaces.

Combining Two Symmetric Forms

The Symmetric Times Symmetric Case

If b and c are both symmetric, the combined form d satisfies

d (v \otimes u, w \otimes x) = b (v, w) \cdot c (u, x) = b (w, v) \cdot c (x, u) = d (w \otimes x, v \otimes u),

so the combined form is again symmetric, since exchanging both pairs of arguments leaves the scalar output unchanged, following directly from the symmetry of each original form applied to its own pair of arguments.

Combining a Symmetric Form with an Alternating Form

The Mixed Case

If b is symmetric while c is alternating, exchanging the arguments of the combined form gives

d (w \otimes x, v \otimes u) = b (w, v) \cdot c (x, u) = b (v, w) \cdot (- c (u, x)) = - d (v \otimes u, w \otimes x),

using the symmetry of b and the antisymmetry of c, so the combined form picks up a sign change under exchange and is therefore alternating whenever exactly one of the two original forms is alternating and the other symmetric.

Combining Two Alternating Forms

Restoration of Symmetry

If both b and c are alternating, applying the same exchange computation produces two sign changes that cancel, giving

d (w \otimes x, v \otimes u) = (- 1) (- 1) d (v \otimes u, w \otimes x) = d (v \otimes u, w \otimes x),

so the combined form is symmetric, matching the standard sign rule that a product of two alternating objects behaves symmetrically, exactly as with a product of two negative numbers producing a positive result.

Summary of the Form Type Relation

The Sign Rule Table

The form type relation follows a rule matching the parity of alternation: symmetric combined with symmetric gives symmetric, symmetric combined with alternating gives alternating, and alternating combined with alternating gives symmetric, so the resulting classification depends only on whether an even or an odd number of the two original forms are alternating.

Applicability to General Forms

If either b or c is a general bilinear form belonging to neither classification, the combined form d likewise generally belongs to neither classification, since no cancellation or reinforcement of the exchange sign is guaranteed without a definite symmetry property to exploit on the relevant factor.