Least change in the Determinant or Permanent of a matrix under perturbation of a single element: continuous and discrete cases

TL;DR

This paper investigates the probability that a matrix's determinant experiences minimal change when a single element is perturbed, considering both continuous and discrete random variable cases and varying sparsity conditions.

Contribution

It introduces a probabilistic framework for assessing determinant stability under element perturbations, distinguishing between continuous and discrete matrix element distributions.

Findings

Probability formulas depend on whether matrix elements are continuous or discrete.

The approach accounts for matrices with varying levels of sparsity.

Provides insights into matrix determinant sensitivity to element changes.

Abstract

We formulate the problem of finding the probability that the determinant of a matrix undergoes the least change upon perturbation of one of its elements, provided that most or all of the elements of the matrix are chosen at random and that the randomly chosen elements have a fixed probability of being non-zero. Also, we show that the procedure for finding the probability that the determinant undergoes the least change depends on whether the random variables for the matrix elements are continuous or discrete.