We will continue our proof of Marcus-Spielman-Srivastava theorem. Last time we end up with showing that the characteristic polynomial of summation of rank 1, positive semi-definite deterministic Hermitian matrices is real and stable via the fact that stable polynomials are closed under restriction and under certain differentiation operators.

This time, we will examine some convexity properties of real stable polynomials. As a result of that, we can show that for summation of rank 1, positive semi-definite random Hermitian matrices, the maximal root of the expected characteristic polynomial is stable and nicely controlled when the expected trace of each summand is bounded.
Speaker(s): Feng Wei (University of Michigan)