Principal Submatrices, Restricted Invertibility, and a Quantitative Gauss–Lucas Theorem

Abstract

Abstract We apply the techniques developed by Marcus, Spielman, and Srivastava, working with principal submatrices in place of rank-$1$ decompositions to give an alternate proof of their results on restricted invertibility. This approach recovers results of theirs’ concerning the existence of well-conditioned column submatrices all the way up to the so-called modified stable rank. All constructions are algorithmic. The main novelty of this approach is that it leads to a new quantitative version of the classical Gauss–Lucas theorem on the critical points of complex polynomials. We show that for any degree $n$ polynomial $p$ and any $c \geq 1/2$, the area of the convex hull of the roots of $p^{(\lfloor cn \rfloor )}$ is at most $4(c-c^2)$ that of the area of the convex hull of the roots of $p$.