Bilinear Forms Intro: Concepts, Formulas and Examples

A bilinear form is a function B(u, v) of two vectors that is linear in each argument separately: doubling the first input doubles the output, and the same holds for the second input. The ordinary dot product is the most familiar example, but bilinear forms also include the skew-symmetric forms that measure signed area.

Bilinear forms encode geometry: inner products give lengths and angles, while general ones describe metrics in relativity and strain in elasticity. Every quadratic form comes from a bilinear form.

Key Concepts

• Bilinearity. B(au1 + bu2, v) = aB(u1, v) + bB(u2, v) and B(u, av1 + bv2) = aB(u, v1) + bB(u, v2) for all scalars a, b.
• Symmetric form. B(u, v) = B(v, u) for all u, v. The dot product is symmetric.
• Skew-symmetric (alternating) form. B(u, v) = −B(v, u), so B(u, u) = 0 for every u.
• Matrix representation. With a basis, B(x, y) = x^T A y where aij = B(ei, ej). Symmetric forms give symmetric matrices.
• Associated quadratic form. Q(x) = B(x, x). For symmetric B, the polarization identity 4B(x, y) = Q(x + y) − Q(x − y) recovers B from Q.

Formulas

• Coordinate formula: B(x, y) = Σ Σ xi aij yj, compactly x^T A y.
• Polarization identity: B(x, y) = (Q(x + y) − Q(x − y)) / 4, for symmetric B.
• Change of basis: the matrix becomes A′ = P^T A P (congruence, not similarity).
• Assumptions and limits: polarization needs a symmetric form; for skew-symmetric B the associated Q is identically zero.

Worked Example

Work with the skew-symmetric form B((x1, x2), (y1, y2)) = x1y2 − x2y1 on R², which gives the signed area of the parallelogram spanned by the inputs.

1. Check bilinearity in the first argument: B((2x1, 2x2), y) = 2x1y2 − 2x2y1 = 2B(x, y). It is linear in the second argument the same way.
2. Check skew-symmetry: B(y, x) = y1x2 − y2x1 = −(x1y2 − x2y1) = −B(x, y).
3. Evaluate at x = (1, 2), y = (3, 4): B(x, y) = 1 × 4 − 2 × 3 = 4 − 6 = −2.
4. Swap the inputs: B(y, x) = 3 × 2 − 4 × 1 = 6 − 4 = 2 = −B(x, y), confirming skew-symmetry numerically.
5. Matrix form: B(x, y) = x^T A y with A = [[0, 1], [−1, 0]], since x^T A y = x1y2 − x2y1. The matrix is skew-symmetric, as expected.

Common Misconceptions

• Confusing bilinear with quadratic forms. Q(x) = B(x, x) only remembers the symmetric part of B; a skew-symmetric form collapses to Q = 0, losing everything.
• Expecting B(x, y) = B(y, x) always. Symmetry is a special property, not a rule; skew-symmetric forms reverse sign on swap.
• Using similarity instead of congruence. Under a change of basis the form's matrix transforms as P^T A P, not P^−1 A P.

Real-World Applications

• Relativity: the spacetime interval is a symmetric bilinear form with signature (+, −, −, −), the Minkowski metric.
• Mechanics: kinetic energy ½v^T M v comes from the symmetric bilinear form defined by the mass matrix M.
• Electromagnetism: the electromagnetic field tensor is a skew-symmetric bilinear form encoding electric and magnetic fields together.

A bilinear form is linear in each of its two inputs, represented in coordinates by x^T A y. Symmetric forms give dot products, quadratic forms, and geometry; skew-symmetric forms give signed areas and the language of electromagnetism and symplectic mechanics.