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We in this paper improve the well-known C. R. Johnson's lower bound for the smallest singular value via permutation matrices. A direct algorithm is also given to compute the new lower bound. [ABSTRACT FROM AUTHOR]

Let R n be an n × n random matrix with i.i.d. subgaussian entries. Let M be an n × n deterministic matrix with norm ‖ M ‖ ≤ n γ where 1 / 2 < γ < 1 . The goal of this paper is to give a general estimate of the smallest singular value of the sum R n + M , which improves an earlier result of Tao and Vu. [ABSTRACT FROM AUTHOR]

We study the problem of super-resolution of a linear combination of Dirac distributions and their derivatives on a one-dimensional circle from noisy Fourier measurements. Following numerous recent works on the subject, we consider the geometric setting of "partial clustering", when some Diracs can be separated much below the Rayleigh limit. Under this assumption, we prove sharp asymptotic bounds for the smallest singular value of a corresponding rectangular confluent Vandermonde matrix with nodes on the unit circle. As a consequence, we derive matching lower and upper min-max error bounds for the above super-resolution problem, under the additional assumption of nodes belonging to a fixed grid. [ABSTRACT FROM AUTHOR]

It is necessary to explore more accurate estimates of the infinity norm of the inverse of a matrix in both theoretical analysis and practical applications. This paper focuses on obtaining a tighter upper bound on the infinite norm of the inverse of Dashnic–Zusmanovich-type (DZT) matrices. The realization of this goal benefits from constructing the scaling matrix of DZT matrices and the diagonal dominant degrees of Schur complements of DZT matrices. The effectiveness and superiority of the obtained bounds are demonstrated through several numerical examples involving random variables. Moreover, a lower bound for the smallest singular value is given by using the proposed bound. [ABSTRACT FROM AUTHOR]

The use of boundary elements in the analysis of exterior acoustic problems poses challenges at specific frequencies, since fictitious eigenfrequencies may arise at the internal resonances of cavities, leading to inaccurate results or even unstable behavior. To filter out these fictitious eigenfrequencies, a scheme based on the combined Helmholtz integral equation formulation (CHIEF) can be used to prevent the so-called non-uniqueness problem, although it requires additional equations and points. The BEM formulation final accuracy will, however, depend on the correct choice of these points. Here, a strategy to help in defining good approximations for the position and number of such points is proposed, based on an optimization process which maximizes the system matrix's smallest singular value. The accuracy of the method for exterior radiation problems is investigated using different examples. With low computational cost and simple implementation, the two proposed algorithms automatically circumvent the non-uniqueness problem, aiding the implementation of more stable BEM codes. [ABSTRACT FROM AUTHOR]

It is necessary to explore more accurate estimates of the infinity norm of the inverse of a matrix in both theoretical analysis and practical applications. This paper focuses on obtaining a tighter upper bound on the infinite norm of the inverse of Dashnic–Zusmanovich-type (DZT) matrices. The realization of this goal benefits from constructing the scaling matrix of DZT matrices and the diagonal dominant degrees of Schur complements of DZT matrices. The effectiveness and superiority of the obtained bounds are demonstrated through several numerical examples involving random variables. Moreover, a lower bound for the smallest singular value is given by using the proposed bound.

Given a real number a ≥ 1, let Kn(a) be the set of all n × n unit lower triangular matrices with each element in the interval [−a, a]. Denoting by λn(·) the smallest eigenvalue of a given matrix, let cn(a) = min {λ n(YYT) : Y ∈ Kn(a)}. Then cn(a)\sqrt {{c_n}\left( a \right)} is the smallest singular value in Kn(a). We find all minimizing matrices. Moreover, we study the asymptotic behavior of cn(a) as n → ∞. Finally, replacing [−a, a] with [a, b], a ≤ 0 < b, we present an open question: Can our results be generalized in this extension?

The use of boundary elements in the analysis of exterior acoustic problems poses challenges at specific frequencies, since fictitious eigenfrequencies may arise at the internal resonances of cavities, leading to inaccurate results or even unstable behavior. To filter out these fictitious eigenfrequencies, a scheme based on the combined Helmholtz integral equation formulation (CHIEF) can be used to prevent the so-called non-uniqueness problem, although it requires additional equations and points. The BEM formulation final accuracy will, however, depend on the correct choice of these points. Here, a strategy to help in defining good approximations for the position and number of such points is proposed, based on an optimization process which maximizes the system matrix’s smallest singular value. The accuracy of the method for exterior radiation problems is investigated using different examples. With low computational cost and simple implementation, the two proposed algorithms automatically circumvent the non-uniqueness problem, aiding the implementation of more stable BEM codes.

We consider three models of sparse random graphs: undirected and directed Erdős–Rényi graphs and random bipartite graph with two equal parts. For such graphs, we show that if the edge connectivity probability p satisfies n p ≥ log n + k (n) with k (n) → ∞ as n → ∞ , then the adjacency matrix is invertible with probability approaching one (n is the number of vertices in the two former cases and the same for each part in the latter case). For n p ≤ log n - k (n) these matrices are invertible with probability approaching zero, as n → ∞ . In the intermediate region, when n p = log n + k (n) , for a bounded sequence k (n) ∈ R , the event Ω 0 that the adjacency matrix has a zero row or a column and its complement both have a non-vanishing probability. For such choices of p our results show that conditioned on the event Ω 0 c the matrices are again invertible with probability tending to one. This shows that the primary reason for the non-invertibility of such matrices is the existence of a zero row or a column. We further derive a bound on the (modified) condition number of these matrices on Ω 0 c , with a large probability, establishing von Neumann's prediction about the condition number up to a factor of n o (1) . [ABSTRACT FROM AUTHOR]

We derive a lower bound on the smallest singular value of a random d-regular matrix, that is, the adjacency matrix of a random d-regular directed graph. Specifically, let C 1 < d < c n / log 2 n and let M n , d be the set of all n × n square matrices with 0 / 1 entries, such that each row and each column of every matrix in M n , d has exactly d ones. Let M be a random matrix uniformly distributed on M n , d . Then the smallest singular value s n (M) of M is greater than n - 6 with probability at least 1 - C 2 log 2 d / d , where c, C 1 , and C 2 are absolute positive constants independent of any other parameters. Analogous estimates are obtained for matrices of the form M - z Id , where Id is the identity matrix and z is a fixed complex number. [ABSTRACT FROM AUTHOR]

Let A = ( a i j ) be a square n × n matrix with i.i.d. zero mean and unit variance entries. It was shown by Rudelson and Vershynin in 2008 that the upper bound for the smallest singular value s n ( A ) is of order n − 1 2 with probability close to one under the additional assumption that the entries of A satisfy E a 11 4 < ∞ . We remove the assumption on the fourth moment and show the upper bound assuming only E a 11 2 = 1 . [ABSTRACT FROM AUTHOR]

We consider a class of sparse random matrices of the form A n = ( ξ i , j δ i , j ) i , j = 1 n , where { ξ i , j } are i.i.d. centered random variables, and { δ i , j } are i.i.d. Bernoulli random variables taking value 1 with probability p n , and prove a quantitative estimate on the smallest singular value for p n = Ω ( log ⁡ n n ) , under a suitable assumption on the spectral norm of the matrices. This establishes the invertibility of a large class of sparse matrices. For p n = Ω ( n − α ) with some α ∈ ( 0 , 1 ) , we deduce that the condition number of A n is of order n with probability tending to one under the optimal moment assumption on { ξ i , j } . This in particular, extends a conjecture of von Neumann about the condition number to sparse random matrices with heavy-tailed entries. In the case that the random variables { ξ i , j } are i.i.d. sub-Gaussian, we further show that a sparse random matrix is singular with probability at most exp ⁡ ( − c n p n ) whenever p n is above the critical threshold p n = Ω ( log ⁡ n n ) . The results also extend to the case when { ξ i , j } have a non-zero mean. We further find quantitative estimates on the smallest singular value of the adjacency matrix of a directed Erdős–Réyni graph whenever its edge connectivity probability is above the critical threshold Ω ( log ⁡ n n ) . [ABSTRACT FROM AUTHOR]

An exchangeable random matrix is a random matrix with distribution invariant under any permutation of the entries. For such random matrices, we show, as the dimension tends to infinity, that the empirical spectral distribution tends to the uniform law on the unit disc. This is an instance of the universality phenomenon known as the circular law, for a model of random matrices with dependent entries, rows, and columns. It is also a non-Hermitian counterpart of a result of Chatterjee on the semi-circular law for random Hermitian matrices with exchangeable entries. The proof relies in particular on a reduction to a simpler model given by a random shuffle of a rigid deterministic matrix, on hermitization, and also on combinatorial concentration of measure and combinatorial Central Limit Theorem. A crucial step is a polynomial bound on the smallest singular value of exchangeable random matrices, which may be of independent interest. [ABSTRACT FROM AUTHOR]

Let { a i j } ( 1 ≤ i , j < ∞ ) be i.i.d. real-valued random variables with zero mean and unit variance and let an integer sequence ( N m ) m = 1 ∞ satisfy m / N m ⟶ z for some z ∈ ( 0 , 1 ) . For each m ∈ N denote by A m the N m × m random matrix ( a i j ) ( 1 ≤ i ≤ N m , 1 ≤ j ≤ m ) and let s m ( A m ) be its smallest singular value. We prove that the sequence ( N m − 1 / 2 s m ( A m ) ) m = 1 ∞ converges to 1 − z almost surely. Our result does not require boundedness of any moments of a i j 's higher than the 2-nd and resolves a long standing question regarding the weakest moment assumptions on the distribution of the entries sufficient for the convergence to hold. [ABSTRACT FROM AUTHOR]

Abstract: Consider the empirical spectral distribution of complex random matrix whose entries are independent and identically distributed random variables with mean zero and variance . In this paper, via applying potential theory in the complex plane and analyzing extreme singular values, we prove that this distribution converges, with probability one, to the uniform distribution over the unit disk in the complex plane, i.e. the well known circular law, under the finite fourth moment assumption on matrix elements. [Copyright &y& Elsevier]

This paper presents a new technique for the reliable computation of the σ-pseudospectrum defined by Λσ( A)={ z∈ C : σmin( A− zI)≤σ} where σmin is the smallest singular value. The proposed algorithm builds an orbit of adjacent equilateral triangles to capture the level curve ϒσ( A)={ z∈ C : σmin( A− zI)=σ} and uses a bisection procedure on specific triangle vertices to compute a numerical approximation to ϒσ. The method is guaranteed to terminate, even in the presence of round-off errors. [ABSTRACT FROM AUTHOR]

Suppose X is an N × n complex matrix whose entries are centered, independent, and identically distributed random variables with variance 1 ∕ n and whose fourth moment is of order O (n − 2). Suppose A is a deterministic matrix whose smallest and largest singular values are bounded below and above respectively, and z ≠ 0 is a complex number. First we consider the matrix X A X ∗ − z , and obtain asymptotic probability bounds for its smallest singular value when N and n diverge to infinity and N ∕ n → γ , 0 < γ < ∞. Then we consider the special case where A = J = 1 i − j = 1 mod n is a circulant matrix. Using the above result, we show that the limit spectral distribution of X J X ∗ exists when N ∕ n → γ , 0 < γ < ∞ and describe the limit explicitly. Assuming that X represents a ℂ N -valued time series which is observed over a time window of length n , the matrix X J X ∗ represents the one-step sample autocovariance matrix of this time series. A whiteness test against an MA correlation model for this time series is introduced based on the above limit result. Numerical simulations show the excellent performance of this test. [ABSTRACT FROM AUTHOR]