  Random butterfly matrices were introduced by Parker in 1995 to remove the
need for pivoting when using Gaussian elimination. The growing applications of
butterfly matrices have often eclipsed the mathematical understanding of how or
why butterfly matrices are able to accomplish these given tasks. To help begin
to close this gap using theoretical and numerical approaches, we explore the
impact on the growth factor of preconditioning a linear system by butterfly
matrices. These results are compared to other common methods found in
randomized numerical linear algebra. In these experiments, we show
preconditioning using butterfly matrices has a more significant dampening
impact on large growth factors than other common preconditioners and a smaller
increase to minimal growth factor systems. Moreover, we are able to determine
the full distribution of the growth factors for a subclass of random butterfly
matrices. Previous results by Trefethen and Schreiber relating to the
distribution of random growth factors were limited to empirical estimates of
the first moment for Ginibre matrices.
