 
Summary: Perturbed identity matrices have high rank: proof and applications
Noga Alon
Abstract
We describe a lower bound for the rank of any real matrix in which all diagonal entries are
significantly larger in absolute value than all other entries, and discuss several applications of
this result to the study of problems in Geometry, Coding Theory, Extremal Finite Set Theory
and Probability. This is partly a survey, containing a unified approach for proving various known
results, but it contains several new results as well.
1 Introduction
Let B = (bi,j) be an n by n real matrix. It is easy and well known that if for every i, bi,i > j=i bi,j,
then B is of full rank. Indeed, assuming this is false, let c = (cj) be a nonzero column vector so that
Bc = 0. Let cr = maxi ci ( > 0) and consider the component number r of Bc. The absolute value
of this component is

j
br,jcj br,rcr 
j=r
br,jcj cr(br,r 
j=r
br,j) > 0,
