Pivot probabilities and norm effects in Gaussian elimination for $\beta$-ensembles

TL;DR

This paper investigates pivot probabilities in Gaussian elimination for random matrix ensembles, resolving discrepancies between theory and practice for GUE matrices and confirming theoretical predictions for beta-ensembles.

Contribution

It derives exact pivot probabilities for GUE matrices under standard implementations and validates theoretical expectations for beta-ensembles.

Findings

Exact pivot probability for GUE matrices derived.

Discrepancy between theory and practice for GUE resolved.

Beta-ensembles agree with theoretical predictions.

Abstract

We analyze pivot probabilities in Gaussian elimination with partial pivoting (GEPP) for $2 \times 2$ random matrix ensembles. For GUE matrices, we resolve a previously reported discrepancy between theoretical predictions and empirical observations by deriving the exact pivot probability under standard LAPACK-style implementations. We further show that Dumitriu-Edelman tridiagonal $\beta$-ensembles agree with the earlier theoretical expectations.