  We use the method of interlacing families of polynomials to derive a simple
proof of Bourgain and Tzafriri's Restricted Invertibility Principle, and then
to sharpen the result in two ways. We show that the stable rank can be replaced
by the Schatten 4-norm stable rank and that tighter bounds hold when the number
of columns in the matrix under consideration does not greatly exceed its number
of rows. Our bounds are derived from an analysis of smallest zeros of Jacobi
and associated Laguerre polynomials.
