Singularity of random Bernoulli matrices. (English) Zbl 1458.15023

Tikhomirov, Konstantin

Ann. Math. (2) 191, No. 2, 593-634 (2020).

Summary: For each \(n\), let \(M_n\) be an \(n\times n\) random matrix with independent \(\pm 1\) entries. We show that \(\mathbb{P}\{M_n\text{ is singular}\}=(1/2+o_n(1))^n\), which settles an old problem. Some generalizations are considered.

MSC:

15A18	Eigenvalues, singular values, and eigenvectors
15B52	Random matrices (algebraic aspects)

Keywords:

Bernoulli matrix; singularity