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Bilinear Forms

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Definition of a Bilinear Form

A bilinear form on a vector space is a function that is linear in each of its two arguments. Given a vector space V over a field F, a bilinear form is a map $B: V \times V \rightarrow F$ such that for all $u, v, w \in V$ and $a \in F$ , $B(u+v, w) = B(u, w) + B(v, w)$ and $B(au, v) = aB(u, v)$ .

Symmetric Bilinear Form

A bilinear form $B$ is symmetric if $B(v, w) = B(w, v)$ for all $v, w \in V$ . Example: The dot product on $\mathbb{R}^n$ , where $B(v, w) = v \cdot w$ .

Alternating Bilinear Form

A bilinear form $B$ is alternating if $B(v, v) = 0$ for all $v \in V$ . Example: The determinant of a $2 \times 2$ matrix when considering the standard bilinear form on $\mathbb{R}^2$ .

Skew-Symmetric Bilinear Form

A bilinear form $B$ is skew-symmetric if $B(v, w) = -B(w, v)$ for all $v, w \in V$ . Example: The standard symplectic form on $\mathbb{R}^{2n}$ .

Matrix Representation of Bilinear Forms

A bilinear form $B$ can be represented by a matrix $M$ such that $B(v, w) = v^T M w$ for column vectors $v, w$ . Example: The matrix representing the dot product in $\mathbb{R}^n$ is the identity matrix $I_n$ .

Change of Basis for Bilinear Forms

Under a change of basis by an invertible matrix $P$ , a bilinear form with matrix $M$ transforms to $M' = P^T M P$ . Example: Changing from the standard basis to another orthonormal basis in $\mathbb{R}^n$ does not change the matrix for the dot product.

Rank of a Bilinear Form

The rank of a bilinear form is the rank of its matrix representation. It represents the maximal number of linearly independent columns (or rows).

Nondegenerate Bilinear Form

A bilinear form is nondegenerate if the only vector $v$ that satisfies $B(v, w) = 0$ for all $w \in V$ is the zero vector. Example: The dot product in $\mathbb{R}^n$ is nondegenerate.

Signature of a Bilinear Form

The signature of a bilinear form is the pair of integers $(p, q)$ , counting the numbers of positive and negative eigenvalues of the associated matrix. Example: The signature of the Minkowski space-time metric is $(3,1)$ .

Kernel of a Bilinear Form

The kernel of a bilinear form $B$ is the set of vectors $v$ such that $B(v, w) = 0$ for all $w \in V$ . For nondegenerate forms, the kernel is trivial, containing only the zero vector.

Orthogonality in Bilinear Forms

Two vectors $v$ and $w$ are orthogonal with respect to a bilinear form $B$ if $B(v, w) = 0$ . Example: In Euclidean space with the dot product, perpendicular vectors are orthogonal.

Positive Definite Bilinear Form

A bilinear form $B$ is positive definite if $B(v, v) > 0$ for all nonzero vectors $v \in V$ . Example: The dot product on $\mathbb{R}^n$ is positive definite.

Induced Norm from a Bilinear Form

A norm can be induced from a positive definite bilinear form $B$ by defining $||v|| = \sqrt{B(v, v)}$ . Example: The standard Euclidean norm arises from the dot product.

Coercivity of a Bilinear Form

A bilinear form $B$ is coercive if there exists $\alpha > 0$ such that $B(v, v) \geq \alpha ||v||^2$ for all $v \in V$ . This implies that $B$ is bounded below by a positive constant times the square of the norm.

Bilinear Form Associated Quadratic Form

A quadratic form $Q$ associated with a bilinear form $B$ is given by $Q(v) = B(v, v)$ . Example: The function $Q(v) = v_1^2 + v_2^2$ for $v \in \mathbb{R}^2$ is the quadratic form associated with the standard dot product.

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