Complex random matrices have no real eigenvalues

Let [Formula: see text] where [Formula: see text] are iid copies of a mean zero, variance one, subgaussian random variable. Let [Formula: see text] be an [Formula: see text] random matrix with entries that are iid copies of [Formula: see text]. We prove that there exists a [Formula: see text] such that the probability that [Formula: see text] has any real eigenvalues is less than [Formula: see text] where [Formula: see text] only depends on the subgaussian moment of [Formula: see text]. The bound is optimal up to the value of the constant [Formula: see text]. The principal component of the proof is an optimal tail bound on the least singular value of matrices of the form [Formula: see text] where [Formula: see text] is a deterministic complex matrix with the condition that [Formula: see text] for some constant [Formula: see text] depending on the subgaussian moment of [Formula: see text]. For this class of random variables, this result improves on the results of Pan–Zhou [Circular law, extreme singular values and potential theory, J. Multivariate Anal. 101(3) (2010) 645–656] and Rudelson–Vershynin [The Littlewood–Offord problem and invertibility of random matrices, Adv. Math. 218(2) (2008) 600–633]. In the proof of the tail bound, we develop an optimal small-ball probability bound for complex random variables that generalizes the Littlewood–Offord theory developed by Tao–Vu [From the Littlewood–Offord problem to the circular law: Universality of the spectral distribution of random matrices, Bull. Amer. Math. Soc.[Formula: see text]N.S.[Formula: see text] 46(3) (2009) 377–396; Inverse Littlewood–Offord theorems and the condition number of random discrete matrices, Ann. of Math.[Formula: see text] 169(2) (2009) 595–632] and Rudelson–Vershynin [The Littlewood–Offord problem and invertibility of random matrices, Adv. Math. 218(2) (2008) 600–633; Smallest singular value of a random rectangular matrix, Comm. Pure Appl. Math. 62(12) (2009) 1707–1739].

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RANDOM MATRICES: THE CIRCULAR LAW

Communications in Contemporary Mathematics ◽

10.1142/s0219199708002788 ◽

2008 ◽

Vol 10 (02) ◽

pp. 261-307 ◽

Cited By ~ 71

Author(s):

TERENCE TAO ◽

VAN VU

Keyword(s):

Strong Convergence ◽

Random Matrices ◽

Random Variable ◽

Singular Value ◽

Additive Combinatorics ◽

Sparse Random Matrices ◽

Circular Law ◽

Bounded Variance ◽

Complex Random Variable ◽

Least Singular Value

Let x be a complex random variable with mean zero and bounded variance σ2. Let Nn be a random matrix of order n with entries being i.i.d. copies of x. Let λ1, …, λn be the eigenvalues of [Formula: see text]. Define the empirical spectral distributionμn of Nn by the formula [Formula: see text] The following well-known conjecture has been open since the 1950's: Circular Law Conjecture: μn converges to the uniform distribution μ∞ over the unit disk as n tends to infinity. We prove this conjecture, with strong convergence, under the slightly stronger assumption that the (2 + η)th-moment of x is bounded, for any η > 0. Our method builds and improves upon earlier work of Girko, Bai, Götze–Tikhomirov, and Pan–Zhou, and also applies for sparse random matrices. The new key ingredient in the paper is a general result about the least singular value of random matrices, which was obtained using tools and ideas from additive combinatorics.

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On minimal singular values of random matrices with correlated entries

Random Matrices Theory and Application ◽

10.1142/s2010326315500069 ◽

2015 ◽

Vol 04 (02) ◽

pp. 1550006 ◽

Cited By ~ 6

Author(s):

F. Götze ◽

A. Naumov ◽

A. Tikhomirov

Keyword(s):

Random Matrices ◽

Random Matrix ◽

Singular Values ◽

Singular Value ◽

The Matrix ◽

Class Of Matrices ◽

Least Singular Value ◽

Mutually Independent

Let X be a random matrix whose pairs of entries Xjk and Xkj are correlated and vectors (Xjk, Xkj), for 1 ≤ j < k ≤ n, are mutually independent. Assume that the diagonal entries are independent from off-diagonal entries as well. We assume that [Formula: see text], for any j, k = 1, …, n and 𝔼 XjkXkj = ρ for 1 ≤ j < k ≤ n. Let Mn be a non-random n × n matrix with ‖Mn‖ ≤ KnQ, for some positive constants K > 0 and Q ≥ 0. Let sn(X + Mn) denote the least singular value of the matrix X + Mn. It is shown that there exist positive constants A and B depending on K, Q, ρ only such that [Formula: see text] As an application of this result we prove the elliptic law for this class of matrices with non-identically distributed correlated entries.

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The strong circular law: A combinatorial view

Random Matrices Theory and Application ◽

10.1142/s2010326321500313 ◽

2020 ◽

pp. 2150031 ◽

Cited By ~ 1

Author(s):

Vishesh Jain

Keyword(s):

Error Rate ◽

Random Matrix ◽

Random Variable ◽

Singular Value ◽

Operator Norm ◽

Additive Combinatorics ◽

Counting Problem ◽

Circular Law ◽

Matrix Formula ◽

Least Singular Value

Let [Formula: see text] be an [Formula: see text] complex random matrix, each of whose entries is an independent copy of a centered complex random variable [Formula: see text] with finite nonzero variance [Formula: see text]. The strong circular law, proved by Tao and Vu, states that almost surely, as [Formula: see text], the empirical spectral distribution of [Formula: see text] converges to the uniform distribution on the unit disc in [Formula: see text]. A crucial ingredient in the proof of Tao and Vu, which uses deep ideas from additive combinatorics, is controlling the lower tail of the least singular value of the random matrix [Formula: see text] (where [Formula: see text] is fixed) with failure probability that is inverse polynomial. In this paper, using a simple and novel approach (in particular, not using machinery from additive combinatorics or any net arguments), we show that for any fixed complex matrix [Formula: see text] with operator norm at most [Formula: see text] and for all [Formula: see text], [Formula: see text] where [Formula: see text] is the least singular value of [Formula: see text] and [Formula: see text] are positive absolute constants. Our result is optimal up to the constants [Formula: see text] and the inverse exponential-type error rate improves upon the inverse polynomial error rate due to Tao and Vu. Our proof relies on the solution to the so-called counting problem in inverse Littlewood–Offord theory, developed by Ferber, Luh, Samotij, and the author, a novel complex anti-concentration inequality, and a “rounding trick” based on controlling the [Formula: see text] operator norm of heavy-tailed random matrices.

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Least singular value, circular law, and Lindeberg exchange

Random Matrices - IAS/Park City Mathematics Series ◽

10.1090/pcms/026/10 ◽

2019 ◽

pp. 461-498

Author(s):

Terence Tao

Keyword(s):

Singular Value ◽

Circular Law ◽

Least Singular Value

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Deriving source-time functions using principal component analysis

Bulletin of the Seismological Society of America ◽

10.1785/bssa0790030711 ◽

1989 ◽

Vol 79 (3) ◽

pp. 711-730

Author(s):

D. W. Vasco

Keyword(s):

Principal Component Analysis ◽

Moment Tensor ◽

Principal Component ◽

Singular Values ◽

Singular Value ◽

Time Function ◽

Basis Functions ◽

Single Source ◽

Source Time Function ◽

The Moment

Abstract Factors such as source complexity, microseismic noise, and lateral heterogeneity all introduce nonuniqueness into the source-time function. The technique of principal component analysis is used to factor the moment tensor into a set of orthogonal source-time functions. This is accomplished through the singular value decomposition of the time-varying moment tensor. The adequacy of assuming a single source-time function may then be examined through the singular values of the decomposition. The F test can also be used to assess the significance of the various principal component basis functions. The set of significant basis functions can be used to test models of the source-time functions, including multiple sources. Application of this technique to the Harzer nuclear explosion indicated that a single source-time function was found to adequately explain the moment tensor. It consists of a single pulse appearing on the diagonal elements of the moment-rate tensor. The decomposition of the moment tensor for a deep teleseism in the Bonin Islands revealed three basis functions associated with relatively large singular values. The F test indicated that only two of the principal components were significant. The principal component associated with the largest singular value consists of a large pulse followed 16-sec later by a diminished pulse. The second principal component, a long-period oscillation, appears to be a manifestation of the poor resolution of the moment-rate tensor at low frequencies.

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Random polytopes obtained by matrices with heavy-tailed entries

Communications in Contemporary Mathematics ◽

10.1142/s0219199719500275 ◽

2019 ◽

Vol 22 (04) ◽

pp. 1950027

Author(s):

O. Guédon ◽

A. E. Litvak ◽

K. Tatarko

Keyword(s):

Compressed Sensing ◽

Convex Hull ◽

Random Matrices ◽

Random Matrix ◽

Singular Value ◽

Small Ball ◽

Random Polytopes ◽

Heavy Tailed ◽

Smallest Singular Value ◽

Quotient Property

Let [Formula: see text] be an [Formula: see text] random matrix with independent entries and such that in each row entries are i.i.d. Assume also that the entries are symmetric, have unit variances, and satisfy a small ball probabilistic estimate uniformly. We investigate properties of the corresponding random polytope [Formula: see text] in [Formula: see text] (the absolute convex hull of rows of [Formula: see text]). In particular, we show that [Formula: see text] where [Formula: see text] depends only on parameters in small ball inequality. This extends results of [A. E. Litvak, A. Pajor, M. Rudelson and N. Tomczak-Jaegermann, Smallest singular value of random matrices and geometry of random polytopes, Adv. Math. 195 (2005) 491–523] and recent results of [F. Krahmer, C. Kummerle and H. Rauhut, A quotient property for matrices with heavy-tailed entries and its application to noise-blind compressed sensing, preprint (2018); arXiv:1806.04261]. This inclusion is equivalent to so-called [Formula: see text]-quotient property and plays an important role in compressed sensing (see [F. Krahmer, C. Kummerle and H. Rauhut, A quotient property for matrices with heavy-tailed entries and its application to noise-blind compressed sensing, preprint (2018); arXiv:1806.04261] and references therein).

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Asymptotic singular value distributions in information theory

10.1093/oxfordhb/9780198744191.013.41 ◽

2018 ◽

Author(s):

Jean-Phillipe Bouchaud ◽

Marc Potters

Keyword(s):

Information Theory ◽

Random Matrices ◽

Matrix Theory ◽

Data Communication ◽

Communication Channels ◽

Singular Values ◽

Singular Value ◽

Model Data ◽

Stieltjes Transform ◽

Empirical Distributions

This article examines asymptotic singular value distributions in information theory, with particular emphasis on some of the main applications of random matrices to the capacity of communication channels. Results on the spectrum of random matrices have been adopted in information theory. Furthermore, information theorists, motivated by certain channel models, have obtained a number of new results in random matrix theory (RMT). Most of those results are related to the asymptotic distribution of the (square of) the singular values of certain random matrices that model data communication channels. The article first provides an overview of three transforms that are useful in expressing the asymptotic spectrum results — Stieltjes transform, η-transform, and Shannon transform — before discussing the main results on the limit of the empirical distributions of the eigenvalues of various random matrices of interest in information theory.

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The Shannon entropy of Sudoku matrices

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2009.0522 ◽

2010 ◽

Vol 466 (2119) ◽

pp. 1957-1975 ◽

Cited By ~ 8

Author(s):

Paul K. Newton ◽

Stephen A. DeSalvo

Keyword(s):

Random Matrices ◽

Relative Entropy ◽

Shannon Entropy ◽

Stochastic Matrix ◽

Singular Values ◽

Latin Squares ◽

Singular Value ◽

Doubly Stochastic Matrix ◽

Doubly Stochastic ◽

Leibler Divergence

We study properties of an ensemble of Sudoku matrices (a special type of doubly stochastic matrix when normalized) using their statistically averaged singular values. The determinants are very nearly Cauchy distributed about the origin. The largest singular value is , while the others decrease approximately linearly. The normalized singular values (obtained by dividing each singular value by the sum of all nine singular values) are then used to calculate the average Shannon entropy of the ensemble, a measure of the distribution of ‘energy’ among the singular modes and interpreted as a measure of the disorder of a typical matrix. We show the Shannon entropy of the ensemble to be 1.7331±0.0002, which is slightly lower than an ensemble of 9×9 Latin squares, but higher than a certain collection of 9×9 random matrices used for comparison. Using the notion of relative entropy or Kullback–Leibler divergence , which gives a measure of how one distribution differs from another, we show that the relative entropy between the ensemble of Sudoku matrices and Latin squares is of the order of 10 −5 . By contrast, the relative entropy between Sudoku matrices and the collection of random matrices has the much higher value, being of the order of 10 −3 , with the Shannon entropy of the Sudoku matrices having better distribution among the modes. We finish by ‘reconstituting’ the ‘average’ Sudoku matrix from its averaged singular components.

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