On the monotonicity of average singular values of complex Gaussian random matrices

Abstract

We prove that the normalized $k$-th moment of the Laguerre unitary ensemble, as a function of the dimension, is decreasing for fixed $k\in(0,1)\cup (2,\infty)$ and increasing for fixed $k\in(1,2).$ As a corollary, this implies that the average singular value of complex Gaussian random matrix decreases with dimension. Conjectured by Bandeira, Kennedy, and Singer (Math. Program., 2016), this property was recently established by Hutnik, for non-negative integer shape parameter $\alpha$. Our result extends this monotonicity to real $\alpha\geq 0$.