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287A: ECE Convex Optimization and Applications Fall 2006 Homework 1 Lecturer: Gert Lanckriet (please indicate whether you are taking the class for letter grade or S/U grade) Due: Friday 10/13/06 (in class) 1. In this problem we examine the geometrical interpretation of the positive de niteness of a matrix. For each of the following cases, determine the shape of the region generated by the constraint xT Ax 1. (a) (2 points) A = (b) (2 points) A = (c) (2 points) A = 2. Let A = AT Rn n . (a) (2 points) Show that tr(A) = n i=1 21 12 1 1 1 1 1 0 0 1 i with i the eigenvalues of A. n i=1 (b) (2 points) Show that |A| = det(A) = i with i the eigenvalues of A. (c) (optional: 2 bonus points) Do the properties in (a) and (b) still hold for a general A Rn n (not necessarily symmetric)? Prove your answer. 3. Consider a partitioned matrix X = XT = AB BT C where A, C are square and symmetric. When C is invertible, we de ne S = A BC 1 B T and call it the Schur complement of C in X. We will now show that X 0 if and only if C 0 and S 0, in two di erent ways: (a) (3 points) Let P = IF with F the same size as B above. What is the 0I relationship between positive-de niteness of X = P XP T and that of X? Find F 0 such that X is block-diagonal, that is, P XP T = , where represents a 0 matrix. Now, complete the proof. 1-1 ECE 287A Homework 1 October 3 Fall 2006 (b) (3 points) Let z = [uT v T ]T . Consider the quadratic form z T Xz. Minimize it with respect to v (you can nd information about di erentiation with respect to a vector in the course reader or the matrix cookbook, posted on the website). The required result should follow immediately. 4. Show the equivalence between the following statements for A = AT Rn n : (a) A 0. (b) (2 points) The eigenvalues of A are positive. (c) (2 points) If Ai represents the submatrix of A formed by taking the rst i rows and i columns of A, then det(Ai ) > 0 for i = 1, . . . , n. (hint: use induction) 5. In this problem, we will prove a result that will be used later during the course. For any pair symmetric of positive semide nite matrices of the same size A, B, tr(AB) 0. To do this, we will proceed as follows: (a) (2 points) Let B = U T 0 00 U , where U is unitary, Rr r is a diagonal matrix and r is the rank of B. Also, de ne: A = that A negative. A11 A12 = U AU T . Show AT A22 12 0 and that the diagonal elements of A11 (aii , i = 1, . . . , r) are non- (b) (2 points) Show that tr(AB) = tr(A11 ). (c) (2 points) Conclude that tr(AB) 0. (d) (optional: 2 bonus points) What can you say if tr(AB) = 0? 6. (2 points) Assume A is an orthogonal matrix. What do we know about its determinant? 7. Which of the following sets are convex? If they are a ne, cones, balls, ellipsoids or polyhedra, say so. Explain your answer. (a) (1 point) A set of the form {x Rn | aT x }. (b) (1 points) A set of the form {x Rn | i xi i , i = 1, . . . , n}. (c) (1 points) A set of the form {x Rn | aT x b1 , aT x b2 }. 1 2 (d) (2 points) The set of points closer to a given point than a given set, i.e., {x | ||x x0 ||2 ||x y||2 , y S} where S Rn . (e) (2 points) The set of points closer to one set than another, i.e., {x | dist(x, S) dist(x, T )} where S, T Rn and dist(x, S) = inf{||x z|| | z S}. (f ) (2 points) The set {a Rk | p(0) = 1, |p(t)| 1 for t } where p(t) = a1 + a2 t + . . . + ak tk 1 . 1-2 ECE 287A Homework 1 October 3 Fall 2006 (g) (2 points) The set {x Rn | ||x a|| 1}. (h) (2 points) The set of positive semide nite matrices of dimension n n. 8. Which of the following sets S are polyhedra? Why (not)? If possible, express S in the form S = {x | Ax b, F x = g}. (a) (2 points) S = {x Rn |x (b) (2 points) S = {x Rn |x 0, xT y 1 for all y with ||y||2 = 1}. 0, xT y 1 for all y with ||y||1 = 1}. 9. (2 points) Show that a set is convex if and only if its intersection with any line is convex. 10. (3 points) Consider the set of rank-k outer products, de ned as {XX T | X Rn k , rank(X) = k}. Describe its conic hull in simple terms. 1-3

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