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.

翻译：我们得到了一个关于复值$d\times d$矩阵$\frac{1}{\sqrt{d}}X$（其元素为独立同分布的$N(0,1)$随机变量）的平均奇异值$\alpha(d)$关于$d$的递推关系，并利用该关系证明$\alpha(d)$随$d$单调递减，收敛至Marchenko-Pastur分布给出的极限。$\alpha(d)$的单调性最近由Bandeira、Kennedy和Singer在研究酉群$\mathcal{U}_d$上的小Grothendieck问题（一个组合优化问题）时提出猜想\cite{BKS}。该结果给出了$\alpha(d)$的精确全局估计、期望最小和最大奇异值的新界，以及期望最大奇异值与期望最小奇异值之比的下界。证明基于与正交多项式Turán行列式理论的联系。我们还讨论了该结果在最初引发猜想问题中的若干应用。