Maximizing a Second-Degree Polynomial on the Unit Sphere

摘要

Let A be a hermitian matrix of order n, and b a known vector in C(n). The problem is to determine which vectors make phi(x) = (x-b)(H)A(x-b) a maximum or minimum on the unit sphere U = (x : x(H)x = 1). The problem is reduced to the determination of a finite point set, the spectrum of (A,b). The theory reduces to the usual theory of hermitian forms when b = O.