14 - Bilinear Forms

Bilinear Form

Let $V$ be a vector space over field $\mathbf{F}$

Bilinear Form on $V$ is a map

$$F:V\times V\to \mathbf{F}$$

such that for all $u,v,w\in V$ and $\lambda \in \mathbf{F}$ then

$$\begin{array}{rlrl}(i)& F(u+v,w)& & =F(u,w)+F(v,w)\\ (ii)& F(u,v+w)& & =F(u,v)+F(u,w)\\ (iii)& F(\lambda v,w)& & =\lambda F(v,w)=F(v,\lambda w)\end{array}$$

Symmetric

$F$ is symmetric if

$$F(v,w)=F(w,v)\text{ for all }v,w\in V$$

Non-Degenerate

$F$ is non-degenerate if

$$F(v,w)=0\text{ for all }v\in V⟹w=0$$

Positive Definite

$F$ is positive definite if

$$F(v,v)>0\text{ for all }v\ne 0\in V$$

Sesquilinear Form

Let $V$ be a vector space over $\mathbb{C}$

Sesquilinear Form on $V$ is a map

$$F:V\times V\to \mathbb{C}$$

such that for all $u,v,w\in V$ and $\lambda \in \mathbb{C}$ then

$$\begin{array}{rlrl}(i)& F(u+v,w)& & =F(u,w)+F(v,w)\\ (ii)& F(u,v+w)& & =F(u,v)+F(u,w)\\ (iii)& F(\overline{\lambda }v,w)& & =\lambda F(v,w)=F(v,\lambda w)\end{array}$$

Conjugate Symmetric

$F$ is symmetric if

$$F(v,w)=F(w,v)\text{ for all }v,w\in V$$