Franklin's Notes

Bilinear form

A bilinear form $F:X\times X\to\mathbb R$ on a vector space $X$ is a function of two variables that is linear in each variable. That is, the partially applied functions obtained by fixing either of the two arguments is a linear function of the other argument. In symbols, There are several conditions used to classify bilinear forms. A bilinear form $F$ on $X$ is
1. symmetric if $F(x,y)=F(y,x)$ for all $x,y\in X$
2. antisymmetric if $F(x,y)=-F(y,x)$ for all $x,y\in X$
3. nondegenerate if $x=0$ is the only value of $x$ such that $F(x,y)=0$ for all $y\in X$
4. positive definite if $F(x,x)>0$ for all $x\ne 0$
5. negative definite if $F(x,x)<0$ for all $x\ne 0$
6. indefinite if it is neither positive nor negative definite
7. a metric tensor if it is symmetric and nondegenerate
8. an inner product if it is a positive or negative definite metric tensor
9. a symplectic struture if it is antisymmetric and nondegenerate

Unlike with general vector spaces, finite-dimensional metric vector spaces can be mapped naturally onto their dual space by sending each $x\in X$ to the map $x^\ast = G(x,-) \in X^\ast$, the linear functional defined by partially applying the metric tensor $G$ to $x$. That is, $x^\ast$ sends each $y\in X$ to $x\cdot y$. In general, if $F$ is any nondegenerate bilinear form on $X$ (not necessarily even a metric tensor), then the map $x\mapsto x^\ast$, denoted $F_{\downarrow}$, is an isomorphism between $X$ and $X^\ast$. (It is simple to check that $F_{\downarrow}$ is linear and has a trivial kernel.) It therefore has an inverse, denoted $F_{\uparrow}$.

Additionally, each non-degenerate bilinear form $F$ on $X$ induces a bilinear form $F^\ast$ on $X^\ast$ defined by which is also nondegenerate and inherits the properties of being symmetric, antisymmetric, nondegenerate, positive/negative definite, or indefinite from $F$.



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