Lower estimates for the singular values of random matrices

Abstract Let Γ be an n × n matrix, whose entries are independent identically distributed (i.i.d.) random variables satisfying the subgaussian tail estimate. We obtain polynomial type lower estimates of the singular numbers of Γ, which hold with probability close to 1. We also show that if A is an N × n matrix with N > n , whose entries are i.i.d. subgaussian random variables, then with high probability the space E = A R n satisfies the conditions of Kashin's theorem, i.e. the l 2 N and l 1 N norms are equivalent on E. Moreover the distance between these norms polynomially depends on δ = ( N − n ) / n . To cite this article: M. Rudelson, C. R. Acad. Sci. Paris, Ser. I 342 (2006).