The singularity probability of random diagonally-dominant Hermitian matrices

TL;DR

This paper characterizes the singularity probability of random diagonally-dominant Hermitian matrices with nonnegative diagonal entries over reals, complex numbers, and quaternions, providing explicit formulas for these probabilities.

Contribution

It explicitly describes the singular locus of such matrices and derives exact formulas for their singularity probabilities, including special cases involving Bernoulli matrices.

Findings

Probability that a perturbed identity matrix is singular: 2^{-(n-1)(n-2)/2}

Probability a symmetric Bernoulli matrix has eigenvalue n: 2^{-(n^2-n+2)/2}

Explicit expressions for singularity probabilities over various fields.

Abstract

In this note we describe the singular locus of diagonally-dominant Hermitian matrices with nonnegative diagonal entries over the reals, the complex numbers, and the quaternions. This yields explicit expressions for the probability that such matrices, chosen at random, are singular. For instance, in the case of the identity $n \times n$ matrix perturbed by a symmetric, zero-diagonal, $\{\pm 1/(n-1)\}$-Bernoulli matrix, this probability turns out to be equal to $2^{-(n-1)(n-2)/2}$. As a corollary, we find that the probability for a $n\times n$ symmetric $\{\pm 1\}$-Bernoulli matrix to have eigenvalue $n$ is $2^{-(n^2-n+2)/2}$.