The average singular value of a complex random matrix decreases with dimension
Abstract
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 MarchenkoPastur 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 Turán determinants of orthogonal polynomials. We also discuss some applications to the problem that originally motivated the conjecture.
 Publication:

arXiv eprints
 Pub Date:
 June 2016
 arXiv:
 arXiv:1606.00494
 Bibcode:
 2016arXiv160600494A
 Keywords:

 Mathematics  Probability;
 Computer Science  Information Theory;
 Mathematics  Classical Analysis and ODEs
 EPrint:
 11 pages