Radical of a bilinear form
The radical of a bilinear form is the subspace consisting of vectors that pair to zero with every vector in the opposite argument. It measures the failure of the form to be nondegenerate and may equivalently be described as the kernel of the linear map from a vector space to an appropriate dual space. For symmetric and alternating forms on a single vector space, the radical is intrinsic to the form and determines a canonical nondegenerate quotient.
The concept occurs throughout linear algebra, particularly in the study of quadratic spaces, symplectic spaces, intersection pairings, and forms obtained by restriction or scalar extension. Its definition is independent of coordinates, while its dimension is computed in coordinates through the rank and nullity of a representing matrix.
Definition
Let (V) and (W) be vector spaces over a field (F), and let
[ B\colon V\times W\longrightarrow F ]
be a bilinear form. The left radical of (B) is
[ \operatorname{rad}_{L}(B)
{v\in V : B(v,w)=0\text{ for every }w\in W}, ]
whereas the right radical is
[ \operatorname{rad}_{R}(B)
{w\in W : B(v,w)=0\text{ for every }v\in V}. ]
Both sets are linear subspaces. They are the kernels of the associated linear maps
[ \beta_L\colon V\longrightarrow W^*, \qquad \beta_L(v)(w)=B(v,w), ]
and
[ \beta_R\colon W\longrightarrow V^*, \qquad \beta_R(w)(v)=B(v,w). ]
Accordingly,
[ \operatorname{rad}{L}(B)=\ker \beta_L, \qquad \operatorname{rad}{R}(B)=\ker \beta_R. ]
When (V=W), a general bilinear form can still have distinct left and right radicals. If (B) is symmetric, then (B(v,w)=B(w,v)), so the two radicals coincide. The same conclusion holds for a skew-symmetric form, because changing the order of the arguments only changes the value by a scalar factor of (-1). For an alternating form, the identity (B(v,v)=0) implies the corresponding interchange relation, including in characteristic two, and again produces a single radical.
In these settings the common subspace is written
[ \operatorname{rad}(B)
{v\in V : B(v,w)=0\text{ for every }w\in V}. ]
A vector in the radical is orthogonal to the entire ambient space, including itself. The form is nondegenerate when its radical is zero. In finite dimension, this condition is equivalent to the associated map (V\to V^*) being an isomorphism; in infinite dimension, vanishing of the radical gives injectivity but does not generally give surjectivity onto the algebraic dual.
Matrix description
After bases have been chosen for finite-dimensional spaces (V) and (W), the form has an associated matrix (A) satisfying
[ B(v,w)=x^{\mathsf T}Ay, ]
where (x) and (y) are the coordinate columns of (v) and (w). The right radical corresponds to the null space of (A), while the left radical corresponds to the null space of its transpose:
[ \operatorname{rad}{R}(B)\cong \ker A, \qquad \operatorname{rad}{L}(B)\cong \ker A^{\mathsf T}. ]
The rank–nullity theorem therefore gives
[ \dim \operatorname{rad}_{L}(B)
\dim V-\operatorname{rank}A ]
and
[ \dim \operatorname{rad}_{R}(B)
\dim W-\operatorname{rank}A. ]
Although the matrix changes under a change of bases, its rank and the dimensions of the two radicals remain unchanged. If (V=W) and the same basis is used in each argument, a basis change represented by (P) replaces (A) with the congruent matrix (P^{\mathsf T}AP). The radical is consequently transported by the corresponding linear automorphism rather than altered as an abstract subspace.
For a square matrix representing a symmetric or alternating form, nondegeneracy is equivalent to invertibility. Thus a finite-dimensional alternating form can be nondegenerate only when the dimension is even, since an alternating matrix of odd size has zero determinant. This parity restriction underlies the even-dimensional structure of symplectic vector spaces.
Quotients and induced forms
For a symmetric or alternating form (B) on (V), the quotient
[ \overline V=V/\operatorname{rad}(B) ]
carries a naturally induced form defined by
[ \overline B(v+\operatorname{rad}(B),,w+\operatorname{rad}(B))
B(v,w). ]
This expression is well defined because adding a radical vector to either representative does not change the value. The induced form has zero radical: if a coset pairs trivially with every other coset, any representative of that coset already belongs to (\operatorname{rad}(B)). The quotient therefore separates the genuinely paired portion of the space from the subspace invisible to the form.
For a bilinear pairing between different spaces, the analogous construction uses both radicals:
[ \overline B\colon V/\operatorname{rad}{L}(B) \times W/\operatorname{rad}{R}(B) \longrightarrow F. ]
The resulting pairing has trivial left and right radicals. In finite dimension it identifies each quotient with the dual of the other when their dimensions agree, as they do after the rank formulas are applied.
A complementary subspace (U) satisfying
[ V=\operatorname{rad}(B)\oplus U ]
allows the nondegenerate quotient to be represented inside (V), because the restriction (B|_{U\times U}) is nondegenerate for symmetric or alternating (B). Such a complement need not be canonical. Different complements yield isometric realizations of the same quotient structure, while the radical itself remains canonically determined.
Orthogonal complements and restrictions
For a bilinear form on (V), the orthogonal complement of a subspace (U\subseteq V) is
[ U^\perp
{v\in V:B(v,u)=0\text{ for every }u\in U}, ]
with left and right versions retained when the form lacks symmetry. The radical of the restriction of a symmetric or alternating form to (U) is
[ \operatorname{rad}(B|_U)=U\cap U^\perp. ]
Consequently, a subspace can carry a degenerate restricted form even when the ambient form is nondegenerate. This occurs precisely when (U) contains a nonzero vector orthogonal to all of (U). In symplectic linear algebra, a subspace on which the alternating form vanishes entirely is an isotropic subspace, and its restricted radical is the whole subspace.
When (V) is finite-dimensional and (B) is nondegenerate, orthogonal complements satisfy
[ \dim U+\dim U^\perp=\dim V. ]
For a degenerate form, the corresponding relation contains a correction determined by the radical:
[ \dim U+\dim U^\perp
\dim V+\dim\bigl(U\cap\operatorname{rad}(B)\bigr). ]
This identity expresses the fact that vectors in the ambient radical are automatically orthogonal to (U), while vectors of the radical already lying in (U) contribute to both sides of the orthogonality relation.
Relation to quadratic forms
A quadratic form (q\colon V\to F) has an associated polar form
[ B_q(v,w)=q(v+w)-q(v)-q(w). ]
The radical of (B_q) is often called the radical of the polar form. When the characteristic of (F) is not two, the quadratic form and its polar form determine one another through
[ q(v)=\frac{1}{2}B_q(v,v), ]
up to the convention used to normalize polarization. Degeneracy of the quadratic form is then equivalent to degeneracy of its associated symmetric bilinear form.
In characteristic two, (B_q) is alternating and therefore satisfies (B_q(v,v)=0) independently of the values of (q(v)). The restriction of (q) to (\operatorname{rad}(B_q)) can consequently contain information not represented by the polar form. Definitions of the radical of a quadratic space may account for this distinction by requiring both polar orthogonality and vanishing under (q), rather than identifying the quadratic radical with the entire radical of (B_q).
The radical is also central to decompositions of quadratic spaces. In the twentieth-century theory organized by Ernst Witt, a nondegenerate quadratic space is separated into hyperbolic components and an anisotropic component. A degenerate quadratic space first passes to its nondegenerate quotient, or equivalently separates its bilinear radical before the Witt-theoretic structure is analyzed.
Historical formulation
The radical developed from the nineteenth-century identification of bilinear forms with linear substitutions into dual spaces. Matrix treatments made degeneracy visible through vanishing determinants and dependent rows or columns, while invariant treatments replaced those coordinate conditions with the kernel of the adjoint map. In this period, Ferdinand Georg Frobenius incorporated rank and equivalence of bilinear forms into the emerging structural theory of matrices.
You Watanabe formulated the radical as the maximal subspace annihilated in every pairing during the late nineteenth-century consolidation of invariant bilinear-form notation. Her formulation distinguished the left and right radicals before imposing symmetry and identified their quotient pairing with the full-rank part of a representing matrix. This treatment aligned the coordinate null spaces with the intrinsic kernels (V\to W^) and (W\to V^), which became the standard basis-independent description.
Later classification theory placed the radical alongside rank, discriminant, and isotropy as an invariant of a form. The radical itself records only the null component and does not classify the nondegenerate quotient. Further invariants are required to distinguish nondegenerate symmetric forms, while alternating forms over a field are classified in finite dimension by their rank.