scholarly journals Least singular value and condition number of a square random matrix with i.i.d. rows

2021 ◽  
pp. 109070
Author(s):  
M. Gregoratti ◽  
D. Maran
Keyword(s):  
Condition Number ◽  
Random Matrix ◽  
Singular Value ◽  
Least Singular Value

scholarly journals On minimal singular values of random matrices with correlated entries

2015 ◽  
Vol 04 (02) ◽  
pp. 1550006 ◽  
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.

scholarly journals The strong circular law: A combinatorial view

2020 ◽  
pp. 2150031 ◽  
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.

scholarly journals Smooth analysis of the condition number and the least singular value

2010 ◽  
Vol 79 (272) ◽  
pp. 2333-2333 ◽  
Author(s):  
Terence Tao ◽  
Van Vu
Keyword(s):  
Condition Number ◽  
Singular Value ◽  
Least Singular Value

On updating the least singular value: a lower bound

CALCOLO ◽  
2003 ◽  
Vol 40 (4) ◽  
pp. 213-229 ◽  
Author(s):  
C. Fassino
Keyword(s):  
Lower Bound ◽  
Singular Value ◽  
Least Singular Value

On computing bounds for the least singular value of a triangular matrix

1975 ◽  
Vol 15 (1) ◽  
pp. 1-4 ◽  
Author(s):  
Ned Anderson ◽  
Ilkka Karasalo
Keyword(s):  
Triangular Matrix ◽  
Singular Value ◽  
Least Singular Value

scholarly journals RANDOM MATRICES: THE CIRCULAR LAW

2008 ◽  
Vol 10 (02) ◽  
pp. 261-307 ◽  
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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