Knowlpedia

Inner product

A positive-definite product on a vector space that defines lengths and angles.

An inner product on a VV over F=R\mathbb{F}=\mathbb{R} or C\mathbb{C} is a map

⟨⋅,⋅⟩:V×V→F\langle\cdot,\cdot\rangle:V\times V\to\mathbb{F}

such that for all u,v,w∈Vu,v,w\in V and a,b∈Fa,b\in\mathbb{F}:

⟨au+bw, v⟩=a⟨u,v⟩+b⟨w,v⟩,⟨u,v⟩=⟨v,u⟩‾,⟨v,v⟩≥0 and ⟨v,v⟩=0  ⟺  v=0.\langle au+bw,\,v\rangle=a\langle u,v\rangle+b\langle w,v\rangle,\qquad \langle u,v\rangle=\overline{\langle v,u\rangle},\qquad \langle v,v\rangle\ge 0\ \text{and }\langle v,v\rangle=0\iff v=0.
Remarks

An inner product is closely related to a (or sesquilinear form in the complex case). It induces a via ∥v∥=⟨v,v⟩\|v\|=\sqrt{\langle v,v\rangle} and defines through the condition ⟨u,v⟩=0\langle u,v\rangle=0.

Examples
  • On Rn\mathbb{R}^n, the standard dot product ⟨x,y⟩=∑i=1nxiyi\langle x,y\rangle=\sum_{i=1}^n x_i y_i is an inner product.
  • On Cn\mathbb{C}^n, the Hermitian product ⟨x,y⟩=∑i=1nxiyi‾\langle x,y\rangle=\sum_{i=1}^n x_i\overline{y_i} is an inner product.
  • If WW is a symmetric positive-definite matrix, then ⟨x,y⟩=xTWy\langle x,y\rangle=x^\mathsf{T}Wy defines an inner product on Rn\mathbb{R}^n.