The asymptotic distribution of the condition number for random circulant matrices

Abstract

AbstractIn this manuscript, we study the limiting distribution for the joint law of the largest and the smallest singular values for random circulant matrices with generating sequence given by independent and identically distributed random elements satisfying the so-called Lyapunov condition. Under an appropriated normalization, the joint law of the extremal singular values converges in distribution, as the matrix dimension tends to infinity, to an independent product of Rayleigh and Gumbel laws. The latter implies that a normalized $$\textit{condition number}$$ condition number  converges in distribution to a Fréchet law as the dimension of the matrix increases.