Condition number of a square matrix with i.i.d. columns drawn from a convex body (Q2880660)

The paper deals with the study of the smallest singular value of square random matrices. Especially, the case when the columns of random matrices are i.i.d. random vectors with an isotropic log-concave distribution is analyzed. Here, a random vector \(X\) in \({\mathbb R}^n\) is said to be log-concave if its distribution has density \(f\) such that for all \(x, y \in {\mathbb R}^n\) and all \(\theta \in (0, 1)\) is \(f((1- \theta)x + \theta y) \geq f(x)^{1 - \theta} f(y)^{\theta}\). The vector \(X\) is called isotropic if it has mean zero and its covariance matrix is the identity. It is proved that the condition number of the matrix with independent columns distributed according to a log-concave isotropic probability measure is of the order of the size of the matrix. From the proof it also follows the estimation about the tail behavior of the condition number.