  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.
