Geometry of the Discriminant Surface for Quadratic Forms

TL;DR

This paper studies the geometric structure of the manifold of quadratic forms with multiple eigenvalues, proving irreducibility and describing its singular points and geometric features for three variables.

Contribution

It establishes the irreducibility of the manifold and characterizes singular points and geometric structure specifically for three-variable quadratic forms.

Findings

The manifold is irreducible.

Scalar matrices are the only singular points for n=3.

The manifold is a straight cylinder over a cone related to a specific orbit.

Abstract

We investigate the manifold $\cal{M}$ of (real) quadratic forms in n > 1 variables having a multiple eigenvalue. In addition to known facts, we prove that 1) $\cal{M}$ is irreducible, 2) in the case of n = 3, scalar matrices and only them are singular points on $\cal{M}$. For $n = 3$, $\cal{M}$ is also described as the straight cylinder over $\cal{M}$$_0$, where $\cal{M}$$_0$ is the cone over the orbit of the diagonal matrix $\diag(1,1,-2)$ by the orthogonal changes of coordinates. We analyze certain properties of this orbit, which occurs a diffeomorphic image of the projective plane.