Bilinear Forms

Bilinear forms generalize the dot product to abstract vector spaces, providing a framework for studying geometric notions like angles, distances, and orthogonality without requiring an inner product structure.

DefinitionBilinear Form

A bilinear form on vector space $V$ over field $\mathbb{F}$ is a function $B: V \times V \to \mathbb{F}$ that is linear in each argument:

$B(c_1\mathbf{u}_1 + c_2\mathbf{u}_2, \mathbf{v}) = c_1B(\mathbf{u}_1, \mathbf{v}) + c_2B(\mathbf{u}_2, \mathbf{v})$
$B(\mathbf{u}, c_1\mathbf{v}_1 + c_2\mathbf{v}_2) = c_1B(\mathbf{u}, \mathbf{v}_1) + c_2B(\mathbf{u}, \mathbf{v}_2)$

In a basis $\{\mathbf{e}_1, \ldots, \mathbf{e}_n\}$ , the bilinear form is represented by matrix $A$ with $a_{ij} = B(\mathbf{e}_i, \mathbf{e}_j)$ : $B(\mathbf{x}, \mathbf{y}) = \mathbf{x}^TA\mathbf{y}$

DefinitionSymmetric and Skew-Symmetric Forms

A bilinear form $B$ is:

Symmetric if $B(\mathbf{u}, \mathbf{v}) = B(\mathbf{v}, \mathbf{u})$ for all $\mathbf{u}, \mathbf{v}$
Skew-symmetric (or antisymmetric) if $B(\mathbf{u}, \mathbf{v}) = -B(\mathbf{v}, \mathbf{u})$
Alternating if $B(\mathbf{v}, \mathbf{v}) = 0$ for all $\mathbf{v}$

The matrix of a symmetric form is symmetric ( $A^T = A$ ); for skew-symmetric forms, $A^T = -A$ .

ExampleExamples of Bilinear Forms

Dot product: $B(\mathbf{x}, \mathbf{y}) = \mathbf{x} \cdot \mathbf{y} = \sum x_iy_i$ (symmetric)
General quadratic: $B(\mathbf{x}, \mathbf{y}) = x_1y_2 - x_2y_1$ (skew-symmetric)
Matrix form: $B(\mathbf{x}, \mathbf{y}) = \mathbf{x}^TA\mathbf{y}$ for any matrix $A$

DefinitionDegenerate and Non-degenerate Forms

A bilinear form $B$ is non-degenerate if for all $\mathbf{v} \neq \mathbf{0}$ , there exists $\mathbf{w}$ with $B(\mathbf{v}, \mathbf{w}) \neq 0$ .

Equivalently, $B$ is non-degenerate if its matrix representation $A$ is invertible.

A form is degenerate if there exists nonzero $\mathbf{v}$ with $B(\mathbf{v}, \mathbf{w}) = 0$ for all $\mathbf{w}$ .

Remark

Unlike inner products, bilinear forms need not be positive definite or even symmetric. They provide a more general algebraic framework. Symmetric bilinear forms relate to quadratic forms, while skew-symmetric forms appear in symplectic geometry and Hamiltonian mechanics. The classification of bilinear forms via canonical matrices generalizes diagonalization theory.