Principal submatrices, restricted invertibility and a quantitative Gauss-Lucas theorem

TL;DR

This paper provides an alternative, algorithmic proof of restricted invertibility using principal submatrices, and introduces a quantitative version of the Gauss-Lucas theorem relating polynomial roots and critical points.

Contribution

It offers a new proof technique for restricted invertibility and establishes a quantitative Gauss-Lucas theorem with bounds on the roots of polynomial derivatives.

Findings

Algorithmic construction of well-conditioned submatrices up to modified stable rank.

Quantitative bounds on the convex hull area of roots of polynomial derivatives.

Extension of classical Gauss-Lucas theorem with explicit area ratio bounds.

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. We show that one can find well conditioned column submatrices all the way upto the so called modified stable rank. All constructions are algorithmic. A byproduct of these results is an interesting 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 $c \geq \frac{1}{2}$, the area of the convex hull of the roots of $p^{(cn)}$ is at most $4(c-c^2)$ that of the area of the convex hull of the roots of $p$.