We obtain a recurrence relation in d for the average singular value α(d) of a complex valued d×d\ matrix d1X with
random i.i.d., N( 0,1) entries, and use it to show that α(d) decreases
monotonically with d to the limit given by the Marchenko-Pastur
distribution.\ The monotonicity of α(d) has been recently conjectured
by Bandeira, Kennedy and Singer in their study of the Little Grothendieck
problem over the unitary group Ud \cite{BKS}, a combinatorial
optimization problem. The result implies sharp global estimates for α(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.Comment: 11 page