2. (Do not hand in) We consider quadratic forms. Let A be an n x n matrix and let X = Consider the function of n variables defined by q(11, 12, . . ., In) = XTAX. . . . In This is called a quadratic form. (a) Show that we may assume that the matrix A in the above definition is symmetric A+A' by proving the following two facts. First, show that is a symmetric 2 matrix. Second, show that XT ( AtAT 2 X = XTAX. (b) We say that A is positive definite if for every vector X # 0, XAX > 0. Assuming A is symmetric, how does being positive definite relate to the eigenvalues of A? (c) Let f(1, 12, . . . , In) be a smooth function of n variables. Recall that the gradient of f, Vf, is the vector whose components are of ari - and that the Hessian matrix of f is the n x n matrix H whose components are the second partial derivatives of f, Hij = arax; Assuming that Xo is a critical point of f, i.e., Vf(Xo) = 0, show that Xo is a local minimum of f if H(Xo) is positive definite and that Xo is a local maximum of f if H(Xo) is negative definite (i.e., if -H(Xo) is positive definite)