36 - Types of Bilinear Forms

Symmetric Bilinear Form

Let $B:V\times V\to \mathbf{F}$ be a bilinear form

$$B\text{ is symmetric }⟺B(v_1,v_2)=B(v_2,v_1)\text{ for all }{v}_1,v_2\in V$$

Positive Definite

Let $V$ be a RELA vector space
Let $B:V\times V\to \mathbb{R}$

$$B\text{ is positive definite}\text{ if }B(v,v)\ge 0\text{ for all }v\in V$$

and

$$B(v,v)=0⟺v=0$$

Note that it is with the same vector!

Conjugate Symmetric

Let $⟨-,-⟩$ be an inner product space

$$⟨v_1,v_2⟩=\overline{⟨v_2,v_1⟩}$$