## SEEMOUS 2016 – Problems

**Problem 1.** Let be a continuous and decreasing real valued function defined on . Prove that

When do we have equality?

**Problem 2.** a) Prove that for every matrix there exists a matrix such that .

b) Prove that there exists a matrix such that for all .

**Problem 3.** Let be idempotent matrices () in . Prove that

where and is the set of matrices with real entries.

**Problem 4.** Let be an integer and set

Prove that

a)

b) .

Some hints follow.

**Hints.** 1. Observe that adding a constant to does not affect the inequality. To solve the first inequality, translate so that it is positive and apply Chebyshev’s integral inequality for a couple of two decreasing functions. A change of variables and the fact that is decreasing allow to conclude. For the second inequality, translate into the negatives and change the sign. Apply the same Chebyshev’s inequality for a positive-decreasing couple of functions. A change of variables allows us again to conclude. Equality occurs if is constant.

2. a) If has simple eigenvalues then it is diagonal and we can use its diagonal decomposition to find . If has double eigenvalues then we use Cayley Hamilton to find .

b) Just take to be a Jordan block.

3. Use Sylvester inequality in the form and the fact that the rank of the product is less than the rank of each matrix. Note that starting with and allows to reduce the problem to matrices. Inductively we only need to show the case which is obvious.

4. a) change the variable .

b) Use a) and note that the series of functions converges uniformly to the the desired result.

A complete solution for Problem 4 can be found here.

Can you please explain exactly how to solve #4?

How does one solve #4 exactly? In particular what is the full solution?

I’ve posted a solution to Problem 4 here