  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 $p$.
