Homework 2

Due: Oct. 15, 14:00:00

Definite matrices

  1. Suppose ARn×n is a real symmetric matrix. Show that A2k is positive semidefinite for all k=1,2,3,.

  2. Let A=(22a242a22).

    • If a=1, compute the eigenvalues of A and show that A is positive definite.
    • If a=2, use Sylvester's criterion to show that A is positive semidefinite.
    • For what real numbers a is matrix A positive semidefinite? You can use any criterion you like.

Convex sets

  1. Show that if C1,C2Rn are two convex sets, so is C1+C2{x1+x2:x1C1,x2C2}.
  2. Let f(x)=Ax+b be an affine function from Rn to Rm, where ARm×n and bRm. Show that if CRm is a convex set, so is its inverse image f1(C){x:f(x)C}.