Symmetric Bilinear Form
Definition
$F$–vector space $V$ Let $B = V \times V \to F$ be a map satisfying the following; it is called a symmetric bilinear form. For $v, u, w \in V$ and $\lambda \in K$,
- $B(v, u) = B(u, v)$
- $B(u + v, w) = B(u, w) + B(v, w)$
- $B(\lambda u, v) = \lambda B(u, v)$
Explanation
Expanded, a symmetric bilinear form is “a function that assigns a single scalar to a pair of vectors, whose scalar value is independent of the order of the two vectors.” In finite dimensions this is the same as symmetric matrices.
Condition 1 is the requirement of being a symmetric function; conditions 2 and 3 are the requirements to be a bilinear form.
Skew-symmetric bilinear form
As the antisymmetric version of a symmetric bilinear form, a skew-symmetric bilinear form is a $B = V \times V \to F$ satisfying the following. For $v, u, w \in V$ and $\lambda \in K$,
- $B(v, u) = -B(u, v)$
- $B(u + v, w) = B(u, w) + B(v, w)$
- $B(\lambda u, v) = \lambda B(u, v)$
In finite dimensions this corresponds to skew-symmetric matrices.
Examples
$n$ dimensional real space $\mathbb{R}^{n}$’s inner product is a canonical example.
$$ \mathbf{x} \cdot \mathbf{y} = x_{1}y_{1} + \cdots + x_{n}y_{n} = y_{1}x_{1} + \cdots + y_{n}x_{n} = \mathbf{y} \cdot \mathbf{x} $$
Similarly, the inner product on the space of real functions also qualifies.
$$ \braket{f, g} = \int f(x)g(x) dx = \int g(x)f(x) dx = \braket{g, f} $$
An inner product, by its definition, is a symmetric bilinear form, but a symmetric bilinear form is not necessarily an inner product. For example, on $\mathbb{R}^{2n}$ define the operation below.
$$ [\mathbf{x}, \mathbf{y}] = x_{1}y_{1} + \cdots + x_{n}y_{n} - x_{n+1}y_{n+1} - \cdots - x_{2n}y_{2n} $$
This is a symmetric bilinear form but not an inner product, because for $\mathbf{x} = \begin{bmatrix}1 & 1 & \cdots & 1 \end{bmatrix}^{\mathsf{T}} \ne \mathbf{0}$ we have $[\mathbf{x}, \mathbf{x}] = 0$.