Suppose is a real symmetric matrix. Show that is

*positive semidefinite*for all .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.

- If , compute the eigenvalues of and show that is

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