Bilinear Form

A bilinear form is a function of two vector variables that is linear in each variable separately. Bilinear forms generalize the dot product and provide the algebraic framework for studying quadratic forms, orthogonality in non-standard metrics, and the duality between a vector space and its dual.

Definition

Definition8.1Bilinear form

Let $V$ be a vector space over a field $F$ . A bilinear form on $V$ is a function $B: V \times V \to F$ satisfying:

$B(\alpha u + \beta v, w) = \alpha B(u, w) + \beta B(v, w)$ (linear in the first argument),
$B(u, \alpha v + \beta w) = \alpha B(u, v) + \beta B(u, w)$ (linear in the second argument).

Definition8.2Symmetric and skew-symmetric

A bilinear form $B$ is:

Symmetric if $B(u, v) = B(v, u)$ for all $u, v$ .
Skew-symmetric (or alternating) if $B(u, v) = -B(v, u)$ for all $u, v$ .

In characteristic $\neq 2$ , every bilinear form decomposes uniquely as $B = B_{\text{sym}} + B_{\text{skew}}$ where $B_{\text{sym}}(u,v) = \frac{1}{2}(B(u,v) + B(v,u))$ and $B_{\text{skew}}(u,v) = \frac{1}{2}(B(u,v) - B(v,u))$ .

Matrix representation

Definition8.3Matrix of a bilinear form

Given a basis $\{e_1, \ldots, e_n\}$ of $V$ , the bilinear form $B$ is represented by the Gram matrix $M = (m_{ij})$ where $m_{ij} = B(e_i, e_j)$ . Then:

$B(u, v) = [u]^T M [v],$

where $[u], [v]$ are the coordinate vectors. $B$ is symmetric iff $M = M^T$ , and skew-symmetric iff $M = -M^T$ .

ExampleThe dot product as a bilinear form

$B(x, y) = x^T y = \sum x_i y_i$ on $\mathbb{R}^n$ . The Gram matrix (in the standard basis) is $M = I_n$ .

$B$ is symmetric, and it is the standard inner product.

ExampleA non-standard symmetric bilinear form

$B(x, y) = x^T \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix} y$ on $\mathbb{R}^2$ .

$B(e_1, e_1) = 2$ , $B(e_1, e_2) = 1$ , $B(e_2, e_2) = 3$ .

$B((1,1), (1,-1)) = (1,1)\begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}\begin{pmatrix} 1 \\ -1 \end{pmatrix} = (1,1)\begin{pmatrix} 1 \\ -2 \end{pmatrix} = -1$ .

ExampleA skew-symmetric bilinear form

$B(x, y) = x_1 y_2 - x_2 y_1$ on $\mathbb{R}^2$ . Gram matrix: $M = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ .

$B(u, v) = -B(v, u)$ ✓. $B(u, u) = 0$ for all $u$ (skew-symmetric forms always satisfy this in characteristic $\neq 2$ ).

Geometrically, $B(u, v) = \det(u, v)$ is the signed area of the parallelogram spanned by $u$ and $v$ .

ExampleThe Lorentz bilinear form

On $\mathbb{R}^4$ : $B(x, y) = -x_0 y_0 + x_1 y_1 + x_2 y_2 + x_3 y_3$ .

Gram matrix: $M = \operatorname{diag}(-1, 1, 1, 1)$ (the Minkowski metric).

This is symmetric but not positive definite -- it has signature $(3, 1)$ . It defines the geometry of special relativity.

Change of basis

Theorem8.1Change of basis for bilinear forms

If $P$ is the change-of-basis matrix from basis $\mathcal{B}$ to $\mathcal{B}'$ , then the Gram matrix transforms as:

$M' = P^T M P.$

Two matrices $M, M'$ related by $M' = P^T M P$ for some invertible $P$ are called congruent.

ExampleDiagonalizing a bilinear form

$M = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ (symmetric, singular).

Complete the square: $B(x, y) = (x_1 + x_2)(y_1 + y_2)$ . Set $u_1 = x_1 + x_2$ , $u_2 = x_2$ , so $P = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ .

$P^T M P = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 2 \\ 1 & 1 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} 4 & 2 \\ 2 & 1 \end{pmatrix}$ ...

Let me redo. Actually, a cleaner approach: eigenvalues of $M$ are $2$ and $0$ . Eigenvectors: $(1,1)$ for $\lambda = 2$ and $(1,-1)$ for $\lambda = 0$ . With $P = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$ : $P^TMP = \begin{pmatrix} 4 & 0 \\ 0 & 0 \end{pmatrix}$ .

In the new basis, $B = 4u_1^2$ (a "degenerate" form with rank $1$ ).

Rank, kernel, and non-degeneracy

Definition8.4Rank and radical

The rank of a bilinear form $B$ is the rank of its Gram matrix $M$ (this is basis-independent since $\operatorname{rank}(P^TMP) = \operatorname{rank}(M)$ for invertible $P$ ).

The radical (or kernel) of $B$ is:

$\operatorname{rad}(B) = \{v \in V : B(v, w) = 0 \text{ for all } w \in V\}.$

$B$ is non-degenerate if $\operatorname{rad}(B) = \{0\}$ , equivalently $\det M \neq 0$ .

ExampleA degenerate bilinear form

$B(x, y) = x_1 y_1$ on $\mathbb{R}^2$ . Gram matrix: $M = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ , rank $1$ .

$\operatorname{rad}(B) = \{(0, t) : t \in \mathbb{R}\}$ (the $y$ -axis). $B$ is degenerate.

ExampleNon-degenerate forms

$B(x, y) = x_1 y_1 - x_2 y_2$ on $\mathbb{R}^2$ . $M = \operatorname{diag}(1, -1)$ , $\det M = -1 \neq 0$ . Non-degenerate.

$B(x, y) = x^T y$ (standard dot product): $M = I$ , $\det = 1$ . Non-degenerate.

The Lorentz form: $\det M = -1 \neq 0$ . Non-degenerate.

Orthogonality with respect to a bilinear form

Definition8.5B-orthogonality

Vectors $u, v$ are $B$ -orthogonal if $B(u, v) = 0$ . For a subspace $W$ :

$W^{\perp_B} = \{v \in V : B(v, w) = 0 \text{ for all } w \in W\}.$

If $B$ is non-degenerate, then $\dim W + \dim W^{\perp_B} = \dim V$ .

ExampleOrthogonality in the Lorentz metric

$B(x, y) = -x_0 y_0 + x_1 y_1$ on $\mathbb{R}^2$ .

$u = (1, 1)$ : $B(u, u) = -1 + 1 = 0$ . This vector is isotropic (self-orthogonal with respect to $B$ , also called a null vector).

$u = (1, 1)$ and $v = (1, -1)$ : $B(u, v) = -1 - 1 = -2 \neq 0$ . Not $B$ -orthogonal.

$u = (1, 0)$ and $v = (0, 1)$ : $B(u, v) = 0$ . These are $B$ -orthogonal, but $B(u, u) = -1 < 0$ (timelike) and $B(v, v) = 1 > 0$ (spacelike).

ExampleIsotropic vectors

A vector $v$ with $B(v, v) = 0$ and $v \neq 0$ is called isotropic (or null). Isotropic vectors exist iff $B$ is indefinite (has both positive and negative eigenvalues, or is degenerate).

For $B(x, y) = x_1 y_2 + x_2 y_1$ : $B(e_1, e_1) = 0$ , $B(e_2, e_2) = 0$ . Both basis vectors are isotropic.

For the standard inner product ( $M = I$ ): there are no nonzero isotropic vectors (positive definiteness).

Sesquilinear forms

Definition8.6Sesquilinear form

On a complex vector space $V$ , a sesquilinear form is a function $H: V \times V \to \mathbb{C}$ that is linear in the first argument and conjugate-linear in the second:

$H(\alpha u, v) = \alpha H(u, v), \quad H(u, \beta v) = \bar{\beta} H(u, v).$

A Hermitian form satisfies additionally $H(u, v) = \overline{H(v, u)}$ . In a basis, $H(u, v) = [u]^* M [v]$ where $M$ is Hermitian ( $M = M^*$ ).

ExampleStandard Hermitian form on C^n

$H(x, y) = x^* y = \sum_i x_i \bar{y}_i$ . Gram matrix: $M = I_n$ .

$H((1, i), (1, -i)) = 1 \cdot 1 + i \cdot i = 1 - 1 = 0$ . These vectors are orthogonal with respect to $H$ .

Summary

RemarkBilinear forms as generalized inner products

Bilinear forms extend the inner product framework:

Positive definite symmetric bilinear forms are inner products.
Indefinite symmetric forms (like the Lorentz metric) arise in physics and geometry.
Skew-symmetric forms (symplectic forms) are fundamental in Hamiltonian mechanics.
The Gram matrix $M$ represents $B$ in a basis, and congruence $P^TMP$ is the equivalence relation.
Non-degeneracy ( $\det M \neq 0$ ) ensures the form induces an isomorphism $V \to V^*$ .
The classification of symmetric bilinear forms (Sylvester's law of inertia) is a central result of this chapter.