math_groups_discgeom

Singular 0/1-matrices, and the hyperplanes spanned by random 0/1-vectors

Thomas Voigt and Günter M. Ziegler— 2006

Let $P(d)$ be the probability that a random 0/1-matrix of size $d \times d$ is singular, and let $E(d)$ be the expected number of 0/1-vectors in the linear subspace spanned by d-1 random independent 0/1-vectors. (So $E(d)$ is the expected number of cube vertices on a random affine hyperplane spanned by vertices of the cube.) We prove that bounds on $P(d)$ are equivalent to bounds on $E(d)$: \[ P(d) = (2^{-d} E(d) + \frac{d^2}{2^{d+1}}) (1 + o(1)). \] We also report about computational experiments pertaining to these numbers.

TitelSingular 0/1-matrices, and the hyperplanes spanned by random 0/1-vectors
VerfasserThomas Voigt and Günter M. Ziegler
Datum2006
Quelle/n
Erschienen inCombinatorics, Probability & Computing, volume 15, pages 463-471
ArtText