The Space of Inner Products

This Demonstration shows the space of triples with , which can be identified with the positive definite quadratic form , together with the orbits of the natural action of the special linear group of matrices with determinant 1. You can vary the quadratic form by changing its three parameters (, , and the discriminant ) and see the point corresponding to the form move within the region of space where the positive definite quadratic forms lie. By varying the matrix parameters you can see the image of the fixed form (which corresponds to another quadratic form with the same discriminant). You also see the part of the orbit of the action contained within the displayed area by checking the "show orbit" checkbox.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


A positive definite quadratic form (or, equivalently, an inner product) on can be identified with a matrix , where . The bilinear form is then . An invertible matrix acts on via the formula , which corresponds to another positive definite quadratic form. If has determinant 1 (i.e. belongs to ), the form has the same discriminant (determinant of the corresponding matrix). Thus the orbits of the action consist of forms with the same value of the discriminant.
[1] K. Hoffman and R. Kunze, Linear Algebra, Englewood Cliffs, NJ: Prentice–Hall, 1971.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+