Skip to content Skip to navigation

Connexions

You are here: Home » Content » Inner Products

Navigation

Content Actions

  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

      What are tags? tag icon

      Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

    • External bookmarks
  • E-mail the authors
  • Rate this module (How does the rating system work?)

    Rating system

    Ratings

    Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

    How to rate a module

    Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

    (0 ratings)

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual Connexions member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

In these lenses

  • richb's DSP display tagshide tags

    This module is included inLens: richb's DSP resources
    By: Richard BaraniukAs a part of collection:"Signals and Systems"

    Comments:

    "My introduction to signal processing course at Rice University."

    Click the "richb's DSP" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

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.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Definition: Inner Product

You may have run across inner products, also called dot products, on n n before in some of your math or science courses. If not, we define the inner product as follows, given we have some xn x n and yn 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=0n1 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 x2 x 2 and y2 y 2 , then we can write the inner product as

<x,y>=xycosθ x y x y θ (2)
where θθ is the angle between xx and yy.

Figure 1: General plot of vectors and angle referred to in above equations.
Figure 1 (inprod_f1.png)

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>=ycosθ x y y θ

Figure 2: Plot of two vectors from above example.
Figure 2 (inprod_f2.png)

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=0cosθ=1 θ 0 θ 1 ).
  • <x,y> x y is zero when cosθ=0θ=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,x0:<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

Comments, questions, feedback, criticisms?

Send feedback