The accumulated distribution of quadratic forms on the sphere Original Research Article

Let A be a real, symmetric matrix, and consider the quadratic form q(v) = vTAv restricted to the unit sphere. Let Q(t) be the area on the unit sphere of the set of vectors v with q(v) < t. In this paper, we study the smoothness properties of Q: it fails to be analytic at the eigenvalues of A, and we give a detailed description of its behavior at such points.