Homework 2

Deadline: Oct. 14, 22:00:00

Definite matrices

  1. Suppose is a real symmetric matrix. Show that is positive semidefinite for all .

  2. Let .

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

Convex sets

  1. Show that an ellipsoid is the image of a ball under a linear transformation.
  2. Verify that the Minkowski additionis a convexity-preserving operation. Namely, show that if are two convex sets, so is .
  3. Let be an affine function from to , where and . Show that if is a convex set, so is its inverse image .