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The average singular value of a complex ...
Abreu, Luís Daniel...

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The average singular value of a complex random matrix decreases with dimension by Abreu, Luís Daniel ( Author )
Lemonjar Open Access
N.A
01-06-2016
We obtain a recurrence relation in d for the average singular value α(d) of a complex valued d×d\ matrix 1d√X 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 Turán determinants of orthogonal polynomials. We also discuss some applications to the problem that originally motivated the conjecture.
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Article
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English
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MYR 0.01
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http://arxiv.org/abs/1606.00494
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