4.19.4 Tensor Bilinear Form Degeneracy Context

Tensor Bilinear Form Degeneracy Context is the collection of conditions and consequences surrounding whether a bilinear form has a nontrivial radical, that is, whether some nonzero vector pairs to zero with every other vector under the form. Degeneracy governs whether a bilinear form can be inverted, whether it induces an isomorphism to a dual space, and how it can be reduced to a non-degenerate form on a smaller space.

Defining Degeneracy

The Radical

For a bilinear form f: V × W → F, the left radical is

and the right radical is defined symmetrically with the roles of v and w exchanged. When V = W and f is symmetric or alternating, the two radicals coincide and are simply called the radical of f. The form is non-degenerate exactly when its radical (and, if distinct, its right radical) is the zero subspace.

Equivalent Characterizations

Non-degeneracy of f is equivalent to the induced linear map V → W*, v ↦ f(v, ·), being injective, and when dim(V) = dim(W) is finite, equivalent to that map being an isomorphism; in matrix terms, relative to any bases, non-degeneracy is equivalent to the representing matrix A being invertible, equivalently det(A) ≠ 0.

Consequences of Degeneracy

Rank Deficiency

If f is degenerate, its rank, the rank of its matrix A, is strictly less than min(dim V, dim W), and the dimension of the radical accounts exactly for the deficiency: dim(V) - rank(f) = dim(rad(f)). Degeneracy is thus quantified precisely, not merely detected, by the size of the radical relative to the ambient dimension.

Failure of the Induced Map to Be Injective

Degeneracy means the map v ↦ f(v, ·) collapses distinct vectors in the radical to the zero functional; the form cannot distinguish a nonzero radical vector from the zero vector using any test against W, so any construction relying on f to separate points of V, such as defining a norm or testing linear independence via the form, fails on radical vectors.

Reduction to a Non-Degenerate Form

Given a degenerate f on V (with V = W), the quotient V / rad(f) carries a naturally induced non-degenerate bilinear form f̄, defined by f̄([v], [v']) = f(v, v'), well defined precisely because f vanishes identically once either argument lies in the radical. Every degenerate form is, in this sense, a non-degenerate form on a smaller quotient space, padded with an extra radical direction contributing nothing to the form's values.

Degeneracy Versus Related but Distinct Notions

Degeneracy Is Not the Same as Indefiniteness

A symmetric bilinear form can be non-degenerate while still failing to be positive or negative definite; the Minkowski form of special relativity, with signature (+,-,-,-), is indefinite yet non-degenerate, since its matrix diag(1,-1,-1,-1) is invertible despite having both positive and negative diagonal entries. Degeneracy specifically concerns invertibility of the associated matrix, not the signs of its eigenvalues.

Degeneracy Is Not the Same as Having Isotropic Vectors

A nonzero vector v with f(v, v) = 0 is called isotropic, but an isotropic vector need not lie in the radical: the light cone vectors of the Minkowski form satisfy f(v,v) = 0 yet pair nontrivially with most other vectors, so they are isotropic without the form being degenerate. Degeneracy requires a vector to pair to zero against every other vector, a strictly stronger condition than merely pairing to zero against itself.

Degeneracy for Asymmetric Pairings

Two Independent Radicals

For a general bilinear pairing f: V × W → F with V ≠ W, the left and right radicals need not coincide and can even have different dimensions if dim(V) ≠ dim(W); a pairing can be left non-degenerate (injective as a map V → W*) while failing to be right non-degenerate, if dim(W) > dim(V), since the induced map W → V* cannot then be surjective and hence cannot be an isomorphism, though it may still be injective.

Perfect Pairings

A pairing is called perfect when both radicals vanish and dim(V) = dim(W) in the finite-dimensional case, making both induced maps V → W* and W → V* isomorphisms simultaneously; the canonical evaluation pairing V* × V → F is always perfect in finite dimensions, providing the standard example against which other pairings' degeneracy is measured.

Degeneracy and the Tensor Product

Degenerate Forms as Non-Full-Rank Tensors

Under the identification of a bilinear form on V × W with an element of V* ⊗ W*, degeneracy corresponds to that tensor having rank strictly less than min(dim V, dim W) when expressed as a sum of elementary tensors of dual vectors; a non-degenerate form corresponds to a tensor of maximal possible rank relative to the smaller of the two spaces, linking the classical notion of matrix rank directly to the notion of tensor rank inherited from the universal property of the tensor product.

Radical as a Kernel Under the Universal Property Correspondence

Since the induced map f̃: V ⊗ W → F (via the universal property) and the induced map v ↦ f(v, ·): V → W* carry equivalent information, the radical of f corresponds to the set of vectors v such that v ⊗ w lies in the kernel of f̃ for every w, connecting the classical linear-algebraic notion of a radical directly to the kernel structure of the linear functional produced by the tensor universal property.

Practical Significance

Testing for Degeneracy

For a finite-dimensional bilinear form, degeneracy is tested by computing the determinant of its matrix in any basis, since congruence transformations A ↦ P^T A Q for invertible P, Q preserve whether det(A) = 0, making the vanishing of the determinant a basis-independent test.

Why Non-Degeneracy Is Frequently Assumed

Many downstream constructions, defining an inverse metric to raise indices, identifying a space with its dual, defining an orthogonal complement with the expected dimension count, require non-degeneracy to work correctly; when a bilinear form of interest turns out to be degenerate, the standard remedy is to pass to the quotient by its radical, recovering a non-degenerate form to which these constructions can then be applied.