Singularity of random Bernoulli matrices.(English)Zbl 1458.15023

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
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