Notes on Classification of Nondegenerate Bilinear Forms

1. Classification of nondegenerate bilinear forms over ℂ

Setting: \(V\) a finite-dimensional vector space over \(\mathbb{C}\), \(B: V\times V \to \mathbb{C}\) bilinear and nondegenerate (matrix \(M\) is invertible).
Equivalence: congruence — \(B'(x,y)=B(Px,Py)\) with \(P\in\mathrm{GL}(V)\); in matrices \(M' = P^{\mathsf T}M P\).

Invariant: The cosquare \(C = M^{-1}M^{\mathsf T}\). Under congruence, \(C \mapsto P^{-1}CP\); also \(C^{\mathsf T} = C^{-1}\), so \(C \in \mathrm{O}_n(\mathbb{C})\).

Result: There are nondegenerate forms that are neither symmetric nor alternating; the symmetric and alternating ones are only two special families.

Complete set of indecomposable canonical blocks (orthogonal direct sum decomposition):

1. Symmetric: \([1]\) (size \(1\times1\), the standard dot product).
2. Alternating (symplectic): \(\begin{pmatrix}0&1\\-1&0\end{pmatrix}\) (size \(2\times2\)).
3. Mixed (generic): For any \(\lambda \in \mathbb{C}\setminus\{0,\pm1\}\) and integer \(m\ge 1\), the \(2m\times2m\) block
   \[ B_{2m}(\lambda) = \begin{pmatrix} 0 & I_m \\ J_m(\lambda) & 0 \end{pmatrix}, \] where \(J_m(\lambda)\) is the \(m\times m\) Jordan block with eigenvalue \(\lambda\).
   Parameter identification: \(B_{2m}(\lambda) \cong B_{2m}(\lambda^{-1})\).
4. Mixed (exceptional): \(B_{2m}(1)\) and \(B_{2m}(-1)\) for \(m\ge 2\). (The \(m=1\) cases are symmetric and alternating, respectively.)

Every nondegenerate bilinear form is congruent to a direct sum of such blocks. The classification is complete: two such sums are congruent iff the blocks match up to permutation and replacing some \(\lambda\) with \(\lambda^{-1}\).

Thus: There is a continuous, infinite family of distinct equivalence classes, parameterized by \(\lambda \in \mathbb{C}\setminus\{0,\pm1\}\) up to inversion.

2. Representation-theoretic viewpoint: orbits in a GL(V)-module

The space of bilinear forms:
\[ \mathcal{B} = V^* \otimes V^*. \] The group \(\mathrm{GL}(V)\) acts by congruence \(g\cdot B = B(g^{-1}\cdot, g^{-1}\cdot)\), which on matrices is \(g^{-\mathsf T} M g^{-1}\). Classifying forms = describing all orbits.

Module decomposition:
\(V^* \otimes V^* = \operatorname{Sym}^2(V^*) \oplus \operatorname{Alt}^2(V^*)\).
Both summands are irreducible polynomial \(\mathrm{GL}(V)\)-modules (highest weights \((2,0,\dots)\) and \((1,1,0,\dots)\)).

Orbit structure: - Symmetric forms (\(\operatorname{Sym}^2\)): Orbits are classified by rank. The nondegenerate ones form a single dense (open) orbit → \(\operatorname{Sym}^2\) is a prehomogeneous vector space. - Alternating forms (\(\operatorname{Alt}^2\)): Orbits also by rank; dense orbit exists iff \(\dim V\) is even (nondegenerate alternating forms). - Whole \(V^*\otimes V^*\): No dense orbit. The generic form has a mixed cosquare with eigenvalues \(\neq \pm1\), leading to a continuous family of orbits with no single generic equivalence class.

Key distinction: Irreducibility of a module does not imply existence of a dense orbit (that’s the prehomogeneous property, which is rare). The sum \(V^*\otimes V^*\) is not prehomogeneous, while its irreducible components are.

3. Parallel with Jordan normal form

Linear operators as elements of \(\mathrm{End}(V) \cong V\otimes V^*\) under conjugation \(X \mapsto gXg^{-1}\).
Bilinear forms as elements of \(V^*\otimes V^*\) under congruence \(B \mapsto g^{-\mathsf T}Bg^{-1}\).

Both are \(\mathrm{GL}(V)\)-orbit problems on tensor products built from the fundamental representation and its dual.

	Endomorphisms	Bilinear forms
Space	\(V\otimes V^*\)	\(V^*\otimes V^*\)
Action	Conjugation	Congruence
Invariants	Characteristic polynomial + Jordan blocks	Cosquare + mixed blocks
Continuous parameter	Eigenvalues \(\lambda\) (up to order)	Pairs \(\lambda, \lambda^{-1}\) (up to order)
Tame classification	Yes (Jordan canonical form)	Yes (Krull–Schmidt categories)

Both are tame representation-theoretic problems: indecomposables form one-parameter families (discrete series + continuous series).

4. Homological / categorical treatment

For operators:
A pair \((V, T)\) is a finitely generated module over \(k[t]\) (a PID). Jordan form arises from the structure theorem for modules over a PID:
\(V \cong \bigoplus k[t]/(t-\lambda)^m\) → a cokernel of a map between free \(k[t]\)-modules (presentation matrix → Smith normal form).

For bilinear forms:
A form \(B: V\times V \to k\) gives a map \(\varphi: V \to V^*\). Congruence corresponds to isomorphism of the pair \((V, \varphi)\).

This pair is equivalent to a self‑adjoint module over the Kronecker algebra \(\Lambda = kQ\) with an anti‑involution. The Kronecker quiver has two vertices and two parallel arrows; self‑adjoint modules encode the map \(\varphi\) and a fixed isomorphism \(V\cong V^{**}\).
\(\Lambda\) is a tame hereditary algebra; its indecomposable modules are classified into preprojectives, preinjectives, and regular modules (parameterized by \(\mathbb{P}^1\)). The self‑adjoint condition selects: - Preprojective/preinjective blocks → symmetric and alternating forms. - Regular blocks → the mixed \(B_{2m}(\lambda)\) (continuous parameter from the regular component).

Thus, just as Jordan form comes from the module theory of a PID, the classification of bilinear forms comes from the module theory of a tame hereditary algebra with a duality.

5. Key takeaways

• There exist nondegenerate bilinear forms over \(\mathbb{C}\) that are neither symmetric nor alternating; they form infinite continuous families classified by the cosquare.
• The problem is naturally the orbit decomposition of \(V^*\otimes V^*\) under \(\mathrm{GL}(V)\).
• While the irreducible submodules \(\operatorname{Sym}^2\) and \(\operatorname{Alt}^2\) have dense orbits (they are prehomogeneous), the full space does not.
• The parallel with Jordan normal form is deep: both are orbit classifications on tensor spaces, and both have a homological formulation (modules over \(k[t]\) vs. self‑adjoint modules over a hereditary algebra).