1. Prove that the set {x : ||Ax + b||2 ≤ c T x + d} is a convex set. 2. Suppose you are given a matrix M ∈ Rn×n . Prove that, or give a counter-example for each of the following. (a) If M ? 0, then for every i, j, Mii ≥ |Mij |. (b) If Mii ≥ |Mij | for for alli, j, and M is symmetric, then M ? 0. (c) If M = P i aia T i where ai ∈ Rn are arbitrary vectors, M ? 0. (d) If M = h M1 M2 M2 M3 i ? 0, then h M1 0 0 M3 i ? 0 where M1, M3 are square block matrices. 3. Recall that hA, Bi = T r(AB) for matrices A and B. Prove that for a symmetric matrix M, M ? 0 if and only if hM, Zi ≥ 0 for all Z ? 0. 4. Consider the set C = {x ∈ R n |x Ax + b x + c ≤ 0} with A ∈ S n , b ∈ R n , and c ∈ R. (a) Show that C is convex if A ? 0. (b) Is the converse of this statement true? (If C is convex, then A ? 0) 5. Show that if S1 and S2 are convex sets in R m×n , then so is their partial sum defined as S = {(x, y1 + y2) | x ∈ R m, y1, y2 ∈ R n , (x, y1) ∈ S1, (x, y2) ∈ S2}. 1
