The singularity probability of random diagonally-dominant Hermitian matrices

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, ${\pm 1/(n-1)}$-Bernoullimatrix,thisprobabilityturnsouttobeequalto$2^{-(n-1)(n-2)/2}$\mathrm{.}Asacorollary,wefindthattheprobabilityfora$n\times n$symmetric${\pm 1}$-Bernoullimatrixtohaveeigenvalue$n$is$2^{-(n^2-n+2)/2}$.