Complex Inner Product

linear-algebra

Definition

Complex Inner Product

Let $V$ be a complex vector space. A complex inner product is an inner product

$$⟨\cdot ,\cdot ⟩:V\times V\to \mathbb{C}$$

satisfying, for all $u,v,w\in V$ and $c\in \mathbb{C}$:

1. Conjugate symmetry:

$$⟨u,v⟩=\overline{⟨v,u⟩}$$

2. Sesquilinearity (linear in the first argument):

$$⟨cu+v,w⟩=c⟨u,w⟩+⟨v,w⟩$$

3. Positive definiteness:

$$⟨v,v⟩\ge 0,⟨v,v⟩=0⟺v=0$$

For ${\mathbb{C}}^n$, the standard complex inner product is

$$⟨x,y⟩=\sum _{i=1}^n\overline{x_i}{y}_i.$$

Einstein Notation: $s=\overline{u_i}{v}_i\in F$

Contrast with the Real Case

In a real inner product space, conjugate symmetry reduces to ordinary symmetry because $\overline{⟨v,u⟩}=⟨v,u⟩$ for $\mathbb{R}$. Sesquilinearity becomes ordinary bilinearity.