The Average Singular Value of a Complex Gaussian Random Matrix Strictly Decreases with Dimension

Abstract

We settle a conjecture of Bandeira, Kennedy and Singer, arising from their analysis of approximation ratios for the little Grothendieck problem over the unitary group, by proving that the average singular value of a normalized square complex Gaussian random matrix strictly decreases with the dimension. The starting point is a recurrence relation of Abreu, derived from the Christoffel--Darboux formula and a Tur\'{a}n determinant for Laguerre polynomials. This recurrence reduces the problem to the estimation of a mixed Laguerre integral. We transform this estimate into an explicit finite inequality by expanding the relevant integrals in the orthogonal basis associated with the weight $x^{1/2}\,\mathrm{e}^{-x}$. A telescoping identity then converts the resulting inequality into a positive form. The final positivity argument combines explicit estimates for central binomial coefficients, a logarithmic lower bound, and a finite verification of small dimensions. This completes the proof that the average singular value strictly decreases with the dimension in the square complex Gaussian case. As a consequence, the same monotonicity is obtained for $N\times(N+\lambda)$ complex Gaussian matrices with fixed rectangularity $\lambda=0,1,2,\dots$.