On the Stationary Values of a Second-Degree Polynomial on the Unit Sphere

(H denotes complex conjugate transpose) is a real number. C. R. Rao of the Indian Statistical Institute, Calcutta, suggested to us the problem of maximizing (or minimizing) 4(x) for complex x on the unit sphere S = { x: xfx = 1 }. Since 4? is a continuous function on the compact set S, such maxima and minima always exist. We here extend the problem to include finding all stationary values of 4?. In summary, our problem is: