Geometry of the Discriminant Surface for Quadratic Forms

Abstract

We investigate the manifold $\mathcal{M}$ of (real) quadratic forms in n > 1 variables having a multiple eigenvalue. In addition to known facts, we prove that 1) $\mathcal{M}$ is irreducible, 2) in the case of n = 3, scalar matrices and only them are singular points on $\mathcal{M}$. For $n=3$, $\mathcal{M}$ is also described as the straight cylinder over $\mathcal{M}$${}_0$, where $\mathcal{M}$${}_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.

26 pages. Cites 9 references. A draft paper. Withdraw earlier versions. Changes since 2009: 1) computer algebra part removed, 2) results from Arnold's book referenced, 3) many points canceled, many points improved