Normal forms for real quadratic forms

TL;DR

This paper studies the structure of non-diagonal normal forms of quadratic forms on R^n, especially for n=3, revealing their geometric properties within the Grassmannian manifold.

Contribution

It characterizes the set of non-diagonal normal forms for quadratic forms on R^3 as the closure of a 5-dimensional submanifold in a Grassmannian.

Findings

Normal forms form a 5-dimensional submanifold

Set of normal forms is closure of a submanifold

Geometric description within Grassmannian

Abstract

We investigate the non-diagonal normal forms of a quadratic form on R^n, in particular for n=3. For this case it is shown that the set of normal forms is the closure of a 5-dimensional submanifold in the 6-dimensional Grassmannian of 2-dimensional subspaces of \R^5.