The Space of Inner Products
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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.
Contributed by: Andrzej Kozlowski (March 2011)
Open content licensed under CC BY-NC-SA
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.
 K. Hoffman and R. Kunze, Linear Algebra, Englewood Cliffs, NJ: Prentice–Hall, 1971.