We obtain a recurrence relation in $d$ for the average singular value $% \alpha (d)$ of a complex valued $d\times d$\ matrix $\frac{ 1}{ \sqrt{ d}}X$ with random i.i.d., N( 0,1) entries, and use it to show that $\alpha (d)$ decreases monotonically with $d$ to the limit given by the Marchenko-Pastur distribution.\ The monotonicity of $\alpha (d)$ has been recently conjectured by Bandeira, Kennedy and Singer in their study of the Little Grothendieck problem over the unitary group $\mathcal{ U}_{ d}$ \cite{ BKS}, a combinatorial optimization problem. The result implies sharp global estimates for $\alpha (d)$, new bounds for the expected minimum and maximum singular values, and a lower bound for the ratio of the expected maximum and the expected minimum singular value. The proof is based on a connection with the theory of Tur\'{ a}n determinants of orthogonal polynomials. We also discuss some applications to the problem that originally motivated the conjecture.
Categories: math.PR cs.IT math.CA math.IT
