Mathematical programming I, 2011: Extra "short" exercises Summary: Mathematical programming I, 2011: Extra "short" exercises Try to give a short and elegant proof. The extra exercises are compulsory and are to be handed in (latest) at the last WC (in groups of max 2 students). Extra exercises due: Mo 9 May, before WC Ex1. Let M Rn×n be a nonsingular lower triangular matrix. Show that M-1 is also lower triangular. Ex2. Let A Rn×n be positive semidefinite and xT Ax = 0. Show that Ax = 0 holds. Ex3. Let A Rn×n be positive definite. Show that A is also nonsingular. Ex4. Show that any eigenvalue of a symmetric matrix A Rn×n must be real. (Euclidian inner product in Cn?) Other exercises to be discussed in the first WC: Ex.2.11,2.13,2.18,2.26. Extra exercises due: Mo 21 May, before WC Ex5. (Ex.4.1) Let Ci Rn, i I (I any index set) be closed, convex sets. Then the set C = iICi is also closed, convex. Ex6. Let · be a norm on Rn and let y Rn be fixed. Show that the function f(x) = y - x is Lipschitz continuous on Rn. Other exercises to be discussed during the second WC: Ex.3.3;3.7;Ex.4.9, Ex.4.11;4.14;4.17;4.19. Extra exercises due: Do 16 Juni, before WC Ex7. Let be given 0 = d0, d2, . . . , dn-1 Rn such that dT j Adi = 0, i = j (A 0). Show that the Collections: Engineering