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Inner Products

Module by: Michael Haag, Justin Romberg

Summary: This module describes the concept of inner products, which leads into our introduction of Hilbert spaces. Examples and properties of both of these concepts are discussed.

Definition: Inner Product

You may have run across inner products, also called dot products, on ℝn n before in some of your math or science courses. If not, we define the inner product as follows, given we have some x∈ℝn x n and y∈ℝn y n
Definition 1: inner product
The inner product is defined mathematically as:
<x,y>=yTx= y 0 y 1 … y n − 1 x 0 x 1 ⋮ x n − 1 =∑i=0n-1 x i y i x y y x y 0 y 1 … y n − 1 x 0 x 1 ⋮ x n − 1 i n 1 0 x i y i (1)

Inner Product in 2-D

If we have x∈ℝ2 x 2 and y∈ℝ2 y 2 , then we can write the inner product as
<x,y>=∥x∥∥y∥cosθ x y x y θ (2)
where θθ is the angle between xx and yy.
inprod_f1.png
Figure 1: General plot of vectors and angle referred to in above equations.
Geometrically, the inner product tells us about the strength of xx in the direction of yy.
Example 1 
For example, if ∥x∥=1 x 1 , then <x,y>=∥y∥cosθ x y y θ
inprod_f2.png
Figure 2: Plot of two vectors from above example.
The following characteristics are revealed by the inner product:
  • <x,y> x y measures the length of the projection of yy onto xx.
  • <x,y> x y is maximum (for given ∥x∥ x , ∥y∥ y ) when xx and yy are in the same direction ( θ=0⇒cosθ=1 θ 0 θ 1 ).
  • <x,y> x y is zero when cosθ=0⇒θ=90° θ 0 θ 90° , i.e. xx and yy are orthogonal.

Inner Product Rules

In general, an inner product on a complex vector space is just a function (taking two vectors and returning a complex number) that satisfies certain rules:
  • Conjugate Symmetry: <x,y>=<x,y>¯ x y x y
  • Scaling: <αx,y>=α<x,y> α x y α x y
  • Additivity: <x+y,z>=<x,z>+<y,z> x y z x z y z
  • "Positivity": ∀x,x≠0:<x,x>>0 x x 0 x x 0
Definition 2: orthogonal
We say that xx and yy are orthogonal if: <x,y>=0 x y 0

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