Bilinear form
Undergraduate
Consider a vector space $V$. The function $B : V \times V \to \reals$ is a bilinear form on $V$ if $B$ is linear in each argument when the other is held fixed. It need not be symmetric or positive.
Examples
Examples
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$(x, y) \mapsto 2 x_1 y_1 + x_2 y_2$ on $\reals^2$
Undergraduate
$(x, y) \mapsto 2 x_1 y_1 + x_2 y_2$ is a bilinear form, because it is linear in $x$ and in $y$.
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$(x, y) \mapsto x_1 y_1 - x_2 y_2$ on $\reals^2$
Undergraduate
$(x, y) \mapsto x_1 y_1 - x_2 y_2$ is a bilinear form.
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$(x, y) \mapsto x_1 y_1 + x_2 y_2$ on $\reals^2$
Undergraduate
$(x, y) \mapsto x_1 y_1 + x_2 y_2$ on $\reals^2$ is an inner product and a bilinear form, because it is the dot product.
Properties
- Every bilinear form on $\reals^n$ has the form $B(x, y) = x^{\top} A y$ for a unique matrix $A$, and $B$ is symmetric if and only if $A$ is.
Distinctions involving this term
- Dot product vs. inner product vs. bilinear form: Bilinear forms, inner products, and the dot product are all ways of multiplying two vectors to get a number, from the most general to the most specific. The dot product is one particular inner product on $\reals^n$. An inner product is a bilinear form that is also symmetric and positive definite.