The Spectra of Large Toeplitz Band Matrices with a Randomly Perturbed Entry

This report is concerned with the union $sp_{\Omega}^{(j,k)}T_{n}(a)$ of all possible spectra that may emerge when perturbing a large $n \times n$ Toeplitz band matrix $T_{n}(a)$ in the $(j,k)$ site by a number randomly chosen from some set $\Omega$. The main results give descriptive bounds and, in several interesting situations, even provide complete identifications of the limit of $sp_{\Omega}^{(j,k)}T_{n}(a)$ as $n \to \infty$. Also discussed are the cases of small and large sets $\Omega$ as well as the "discontinuity of the infinite volume case", which means that in general $sp_{\Omega}^{(j,k)}T_{n}(a)$ does not converge to something close to $sp_{\Omega}^{(j,k)}T_{n}(a)$ as $n \to \infty$, where $T(a)$ is the corresponding infinite Toeplitz matrix. Illustrations are provided for tridiagonal Toeplitz matrices, a notable special case. \ud \ud The second author was supported by UK Enginering and Physical Sciences Research Council Grant GR/M1241

On the Spectra and Pseudospectra of a Class of Non-Self-Adjoint Random Matrices and Operators

In this paper we develop and apply methods for the spectral analysis of non-self-adjoint tridiagonal infinite and finite random matrices, and for the spectral analysis of analogous deterministic matrices which are pseudo-ergodic in the sense of E.B.Davies (Commun. Math. Phys. 216 (2001), 687-704). As a major application to illustrate our methods we focus on the "hopping sign model" introduced by J.Feinberg and A.Zee (Phys. Rev. E 59 (1999), 6433-6443), in which the main objects of study are random tridiagonal matrices which have zeros on the main diagonal and random $\pm 1$'s as the other entries. We explore the relationship between spectral sets in the finite and infinite matrix cases, and between the semi-infinite and bi-infinite matrix cases, for example showing that the numerical range and $p$-norm \eps-pseudospectra (\eps>0, $p\in [1,\infty]$) of the random finite matrices converge almost surely to their infinite matrix counterparts, and that the finite matrix spectra are contained in the infinite matrix spectrum $\Sigma$. We also propose a sequence of inclusion sets for $\Sigma$ which we show is convergent to $\Sigma$, with the $n$th element of the sequence computable by calculating smallest singular values of (large numbers of) $n\times n$ matrices. We propose similar convergent approximations for the 2-norm \eps-pseudospectra of the infinite random matrices, these approximations sandwiching the infinite matrix pseudospectra from above and below

Coburn's lemma and the finite section method for random Jacobi operators

We study the spectra and pseudospectra of semi-infinite and bi-infinite tridiagonal random matrices and their finite principal submatrices, in the case where each of the three diagonals varies over a separate compact set, say U,V,W\subset\C. Such matrices are sometimes termed stochastic Toeplitz matrices $A_+$ in the semi-infinite case and stochastic Laurent matrices $A$ in the bi-infinite case. Their spectra, \S=\spec A and \S_+=\spec A_+, are independent of $A$ and $A_+$ as long as $A$ and $A_+$ are pseudoergodic (in the sense of E.B.~Davies, {\em Commun.~Math.~Phys.} {\bf 216} (2001), 687--704), which holds almost surely in the random case. This was shown in Davies (2001) for $A$; that the same holds for $A_+$ is one main result of this paper. Although the computation of $\S$ and $\S_+$ in terms of $U$, $V$ and $W$ is intrinsically difficult, we give upper and lower spectral bounds, and we explicitly compute a set $G$ that fills the gap between $\S$ and $\S_+$ in the sense that $\S\cup G=\S_+$. We also show that the invertibility of one (and hence all) operators $A_+$ implies the invertibility -- and uniform boundedness of the inverses -- of all finite tridiagonal square matrices with diagonals varying over $U$, $V$ and $W$. This implies that the so-called finite section method for the approximate solution of a system $A_+x=b$ is applicable as soon as $A_+$ is invertible, and that the finite section method for estimating the spectrum of $A_+$ does not suffer from spectral pollution. Both results illustrate that tridiagonal stochastic Toeplitz operators share important properties of (classical) Toeplitz operators. Indeed, one of our main tools is a new stochastic version of the Coburn lemma for classical Toeplitz operators, saying that a stochastic tridiagonal Toeplitz operator, if Fredholm, is always injective or surjective. In the final part of the paper we bound and compare the norms, and the norms of inverses, of bi-infinite, semi-infinite and finite tridiagonal matrices over $U$, $V$ and $W$. This, in particular, allows the study of the resolvent norms, and hence the pseudospectra, of these operators and matrices

The Non-Hermitian Skin Effect With Three-Dimensional Long-Range Coupling

We study the non-Hermitian skin effect in a three-dimensional system of finitely many subwavelength resonators with an imaginary gauge potential. We introduce a discrete approximation of the eigenmodes and eigenfrequencies of the system in terms of the eigenvectors and eigenvalues of the so-called gauge capacitance matrix $\mathcal{C}_N^\gamma$, which is a dense matrix due to long-range interactions in the system. Based on translational invariance of this matrix and the decay of its off-diagonal entries, we prove the condensation of the eigenmodes at one edge of the structure by showing the exponential decay of its pseudo-eigenvectors. In particular, we consider a range-k approximation to keep the long-range interaction to a certain extent, thus obtaining a k-banded gauge capacitance matrix $\mathcal{C}_{N,k}^\gamma$ . Using techniques for Toeplitz matrices and operators, we establish the exponential decay of the pseudo-eigenvectors of $\mathcal{C}_{N,k}^\gamma$ and demonstrate that they approximate those of the gauge capacitance matrix $\mathcal{C}_N^\gamma$ well. Our results are numerically verified. In particular, we show that long-range interactions affect only the first eigenmodes in the system. As a result, a tridiagonal approximation of the gauge capacitance matrix, similar to the nearest-neighbour approximation in quantum mechanics, provides a good approximation for the higher modes. Moreover, we also illustrate numerically the behaviour of the eigenmodes and the stability of the non-Hermitian skin effect with respect to disorder in a variety of three-dimensional structures.Comment: 32 pages, 11 figure

Sparsity-Cognizant Total Least-Squares for Perturbed Compressive Sampling

Solving linear regression problems based on the total least-squares (TLS) criterion has well-documented merits in various applications, where perturbations appear both in the data vector as well as in the regression matrix. However, existing TLS approaches do not account for sparsity possibly present in the unknown vector of regression coefficients. On the other hand, sparsity is the key attribute exploited by modern compressive sampling and variable selection approaches to linear regression, which include noise in the data, but do not account for perturbations in the regression matrix. The present paper fills this gap by formulating and solving TLS optimization problems under sparsity constraints. Near-optimum and reduced-complexity suboptimum sparse (S-) TLS algorithms are developed to address the perturbed compressive sampling (and the related dictionary learning) challenge, when there is a mismatch between the true and adopted bases over which the unknown vector is sparse. The novel S-TLS schemes also allow for perturbations in the regression matrix of the least-absolute selection and shrinkage selection operator (Lasso), and endow TLS approaches with ability to cope with sparse, under-determined "errors-in-variables" models. Interesting generalizations can further exploit prior knowledge on the perturbations to obtain novel weighted and structured S-TLS solvers. Analysis and simulations demonstrate the practical impact of S-TLS in calibrating the mismatch effects of contemporary grid-based approaches to cognitive radio sensing, and robust direction-of-arrival estimation using antenna arrays.Comment: 30 pages, 10 figures, submitted to IEEE Transactions on Signal Processin

Pseudoergodic operators and periodic boundary conditions

There is an increasing literature on random non self-adjoint infinite matrices with motivations ranging from condensed matter physics to neural networks. Many of these operators fall into the class of ``pseudoergodic'' operators, which allows the elimination of probabilistic arguments when studying spectral properties. Parallel to this is the increased awareness that spectral properties of non self-adjoint operators, in particular stability, may be better captured via the notion of pseudospectra as opposed to spectra. Although it is well known that the finite section method applied to these matrices does not converge to the spectrum, it is often found in practice that the pseudospectrum behaves better with appropriate boundary conditions. We make this precise by giving a simple proof that the finite section method with periodic boundary conditions converges to the pseudospectrum of the full operator. Our results hold in any dimension (not just for banded bi-infinite matrices) and can be considered as a generalisation of the well known classical result for banded Laurent operators and their circulant approximations. Furthermore, we numerically demonstrate a convergent algorithm for the pseudospectrum including cases where periodic boundary conditions converge faster than the method of uneven sections. Finally we show that the result carries over to pseudoergodic operators acting on $l^p$ spaces for $p\in[1,\infty]$.This work was supported by EPSRC grant EP/L016516/1

On the performance of minimum redundancy array for multisource direction finding

As an application of power spectrum estimation, the multi-source direction finding has been evolved from conventional FFT method to Superresolution methods such as Multiple Signal Classification(MUSIC) algorithm. Uniform Regular Array(URA) was mainly used in all these approaches. The Minimum Redundancy array(MRA); a non-uniform thinned array which results in an input signals covariance matrix with minimum redundancy has been shown to have certain interesting properties for spectrum estimation. Only recently it was suggested to use the MRA for spatial estimation. The purpose of this research was to study the performance of this array in multi-source direction finding estimation and compare it to the result obtained with URA. Although the emphasis in this research is on using the popular MUSIC algorithm, other algorithms are also considered. Among the topics related to the MRA performance studied in the course of this research are 1. Effect of random displacement of the array element location on the performance of multi-source direction finding. 2. Performance of the MRA versus the URA using MUSIC and Minimum-Norm algorithms. 3. Performance of the MUSIC based direction finding using different covariance matrix estimates for URA and MRA. 4. The error probability of estimating the number (two in particular) of closely located sources with MRA versus URA