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Convex Analysisの二,三の進展について

The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and Toeplitz matrices with absolutely summable elements are derived in a tutorial manner. Mathematical elegance and generality are sacrificed for conceptual simplicity and insight in the hope of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject. By limiting the generality of the matrices considered, the essential ideas and results can be conveyed in a more intuitive manner without the mathematical machinery required for the most general cases. As an application the results are applied to the study of the covariance matrices and their factors of linear models of discrete time random processes.

/pdf/toeplitz-and-circulant-matrices-3e6lj1a9jc.pdf

Toeplitz and circulant matrices

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such ``area laws'' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states.

Colloquium: Area laws for the entanglement entropy

Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

The paper is concerned with finite Hermitian Toeplitz matrices whose entries in the first row grow like a polynomial. Such matrices cannot be viewed as truncations of an infinite Toeplitz matrix which is generated by an integrable function or a nice measure. The main results describe the first-order asymptotics of the extreme eigenvalues as the matrix dimension goes to infinity and also deliver unexpected barriers for the eigenvalues. One purpose of the paper is to popularize once more that questions on the eigenvalues of matrices can be answered in a very elegant way by passing to integral operators. This idea was introduced by Harold Widom about fifty years ago. In this way one can also give an alternative proof to results by William F. Trench on Hermitian Toeplitz matrices with increasing entries.

https://www.math.cinvestav.mx/%7Egrudsky/Papers/118_29062012_Albrecht.pdf

Eigenvalues of Hermitian Toeplitz matrices with polynomially increasing entries

Given a real or complex $n \times n$ matrix $A_n$, we compute the expected value and the variance of the random variable $\| A_n x\|^2/\| A_n \|^2$, where $x$ is uniformly distributed on the unit sphere of $R^n$ or $C^n$. The result is applied to several classes of structured matrices. It is in particular shown that if $A_n$ is a Toeplitz matrix $T_n(b)$, then for large $n$ the values of $\| A_n x\|/\| A_n \|$ cluster fairly sharply around $\| b \|_2/\| b \|_\infty$ if $b$ is bounded and around zero in case $b$ is unbounded.

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The Norm of the Product of a Large Matrix and a Random Vector

A Continuous Analogue of the Fisher-Hartwig Formula for Piecewise Continuous Symbols

We describe the asymptotics of the spectral norm of finite Toeplitz matrices generated by functions with Fisher-Hartwig singularities as the matrix dimension goes to infinity. In the case of positive generating functions, our result provides the asymptotics of the largest eigenvalue, which is of interest in time series with long range memory.

Albrecht Böttcher

Papers

Eigenvalues of Hermitian Toeplitz matrices with polynomially increasing entries

The Norm of the Product of a Large Matrix and a Random Vector

A Continuous Analogue of the Fisher-Hartwig Formula for Piecewise Continuous Symbols

Norms of Toeplitz Matrices with Fisher-Hartwig Symbols

Infinite Toeplitz and Laurent matrices with localized impurities