  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.
