Maximizing a Second-degree Polynomial on the Unit Sphere

Let A be a hermitian matrix of order n, and b a known vector in n C . The problem is to determine which vectors make 0(x) = (x-b)H A(x-b) a maximum or minimum on the unit sphere U = [x H : xx= 13 l 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 = 0. 1* Reproduction in Whole or in Part is Permitted for any Purpose of the United States Government. This report was supported in part by Office of Naval Research Contract Nonr-225(37) (NR-044-211) at Stanford University. 1. The problem. Let A be a hermitian square matrix of complex elements and order n. Let b be a known n-vector of complex numbers. For each complex n-vector x, the nonhomogeneous quadratic expression OJ) o(x) = (x-b)HA(x-b) (H denotes complex conjugate transpose) is a real number. The problem, suggested to us by C. R. Rao of the Indian Statistical Institute, Calcutta, is to maximize (or minimize) o(x) for complex x on the unit sphere s {x: xHx = = 1). Since 0 is a continuous function on the compact set S, such maxima and minima always exist. In summary, our problem is: O-2) maximize or minimize a(x) subject to xHx =l . The purpose of this note is to reduce the problem (1.2) to the determination of a certain finite real point set which we shall call the spectrum of the system (A,b) (defined at end of Sec. l), and show that a unique number h in the spectrum determines the one or more x = xh which maximize Q(x) for given b. Theorem (4.1) is the main result. The development is an extension to general b of the familiar theory for the homogeneous case when b = 8, the zero vector. No consideration to a practical computer algorithm is given here. In Sec. 7 we show that determining the least-squares solution of an arbitrary system of linear equations Cy = f, subject to the quadratic 1 constraint yHY = 1, is a special case of problem (1.2). Phillips (9.2) and Twomey (9.3) begin the actual numerical solution of certain integral equations by approximating them with algebraic problems very closely related to the minimum problem (1.2). ⌧1 5 ⌧2 < l l l < ⌧n be the (necessarily real) eigenvalues of A, by..,Un3 be a corresponding real orthonormal set of eigenvectors, = X+u; (i=l,...,n). Let and let with Aui I L 103) Let a given b be written n b = c i=l biui . (1.4) Theorem. If x is any vector in S for which o(x) is stationary with respect to Sg then there exists a real number X = X(x) such that (105) (1.6) Conversely, if any real X renders Q(x) stationary. A(x-b) = Xx , H xx=1 . X and vector x satisfy (1.5, 1.6), then Proof. Let xo be a point of S. Now, as shown in lemma (8.7), Q(x) is stationary at x0 with respect to x in S, if and only if there Y exists a real Lagrange multiplier X such that q(x) = (x-b)HA(x-b) h xHx is stationary at x0 with respect to all neighboring complex vectors x. Since 0 = -& grad q(x,) = A(xo-b) X x 0 ’ the theorem is proved. 2 To see what conditions are satisfied by the X of theorem (1.4), we note that the system (1.5, 1.6) is equivalent to the system (1.7) (A-hI)x = Ab , om H xx=1 . Let n X= c X.U. l i=l 1 1 Then (1.7) is equivalent to 0-Y) f (Xi-X)xiui = i hibiui . i=l i=l Definition. By the spectrum of the pair (A,b) we mean the set of all real X for which there exists an x such that (1.7) and (1.8) are satisfied. Given any X, x satisfying (1.7) and (1.8), we shall say that x belongs to X, and frequently write x x in the form x . Note that the spectrum of (A& is the ordinary spectrum (Xi] of A. 2. Special case: no Xibi = 0. Assume for the present section that Xibi # 0 (all i). This implies that all Xi # 0, i.e., that A is nonsingular. If X is in the spectrum of (A,b), (1.9) implies that X # Xi for all i, and also that