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Gaussian elimination with partial pivoting (GEPP) is a widely used method tosolve dense linear systems. Each GEPP step uses a row transposition pivotmovement if needed to ensure the leading pivot entry is maximal in magnitudefor the leading column of the remaining untriangularized subsystem. We will usetheoretical and numerical approaches to study how often this pivot movement isneeded. We provide full distributional descriptions for the number of pivotmovements needed using GEPP using particular Haar random ensembles, as well ascompare these models to other common transformations from randomized numericallinear algebra. Additionally, we introduce new random ensembles with fixedpivot movement counts and fixed sparsity, $\alpha$. Experiments estimating theempirical spectral density (ESD) of these random ensembles leads to a newconjecture on a universality class of random matrices with fixed sparsity whosescaled ESD converges to a measure on the complex unit disk that depends on$\alpha$ and is an interpolation of the uniform measure on the unit disk andthe Dirac measure at the origin.

Distribution of the number of pivots needed using Gaussian elimination with partial pivoting on random matrices