Inner product

Definition

Standard inner product is defined as following for two vectors $x,y\in C^n$ $⟨x,y⟩=x{y}^{\ast }$ Observe that $x\in C^{1\times n}$ and $y\in C^{n\times 1}$. Then the result is a single complex number. $⟨x,y⟩={\sum }_{i=1}^n{x}_i{\bar{y}}_i$

Properties

• $⟨x,y⟩=\overline{⟨y,x⟩}$
• $⟨x,y+z⟩=⟨x,y⟩+⟨x,z⟩$
• $⟨cx,y⟩=c⟨x,y⟩$
• $⟨x,cy⟩=\bar{c}⟨x,y⟩$
• $⟨x,x⟩\ge 0$
• $⟨x,x⟩=0⟺x=0$