 
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 M1 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, . . . , dn1 Rn such that dT
j Adi = 0, i = j (A 0). Show that the
