Random Circulant Matrices

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Publisher : CRC Press
ISBN 13 : 0429788193
Total Pages : 192 pages
Book Rating : 4.4/5 (297 download)

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Book Synopsis Random Circulant Matrices by : Arup Bose

Download or read book Random Circulant Matrices written by Arup Bose and published by CRC Press. This book was released on 2018-11-05 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt: Circulant matrices have been around for a long time and have been extensively used in many scientific areas. This book studies the properties of the eigenvalues for various types of circulant matrices, such as the usual circulant, the reverse circulant, and the k-circulant when the dimension of the matrices grow and the entries are random. In particular, the behavior of the spectral distribution, of the spectral radius and of the appropriate point processes are developed systematically using the method of moments and the various powerful normal approximation results. This behavior varies according as the entries are independent, are from a linear process, and are light- or heavy-tailed. Arup Bose obtained his B.Stat., M.Stat. and Ph.D. degrees from the Indian Statistical Institute. He has been on its faculty at the Theoretical Statistics and Mathematics Unit, Kolkata, India since 1991. He is a Fellow of the Institute of Mathematical Statistics, and of all three national science academies of India. He is a recipient of the S.S. Bhatnagar Prize and the C.R. Rao Award. He is the author of three books: Patterned Random Matrices, Large Covariance and Autocovariance Matrices (with Monika Bhattacharjee) and U-Statistics, M_m-Estimators and Resampling (with Snigdhansu Chatterjee). Koushik Saha obtained a B.Sc. in Mathematics from Ramakrishna Mission Vidyamandiara, Belur and an M.Sc. in Mathematics from Indian Institute of Technology Bombay. He obtained his Ph.D. degree from the Indian Statistical Institute under the supervision of Arup Bose. His thesis on circulant matrices received high praise from the reviewers. He has been on the faculty of the Department of Mathematics, Indian Institute of Technology Bombay since 2014.

Toeplitz and Circulant Matrices

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Author :
Publisher : Now Publishers Inc
ISBN 13 : 1933019239
Total Pages : 105 pages
Book Rating : 4.9/5 (33 download)

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Book Synopsis Toeplitz and Circulant Matrices by : Robert M. Gray

Download or read book Toeplitz and Circulant Matrices written by Robert M. Gray and published by Now Publishers Inc. This book was released on 2006 with total page 105 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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. 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.

Random Matrices and Non-Commutative Probability

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Publisher : CRC Press
ISBN 13 : 1000458814
Total Pages : 287 pages
Book Rating : 4.0/5 (4 download)

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Book Synopsis Random Matrices and Non-Commutative Probability by : Arup Bose

Download or read book Random Matrices and Non-Commutative Probability written by Arup Bose and published by CRC Press. This book was released on 2021-10-26 with total page 287 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is an introductory book on Non-Commutative Probability or Free Probability and Large Dimensional Random Matrices. Basic concepts of free probability are introduced by analogy with classical probability in a lucid and quick manner. It then develops the results on the convergence of large dimensional random matrices, with a special focus on the interesting connections to free probability. The book assumes almost no prerequisite for the most part. However, familiarity with the basic convergence concepts in probability and a bit of mathematical maturity will be helpful. Combinatorial properties of non-crossing partitions, including the Möbius function play a central role in introducing free probability. Free independence is defined via free cumulants in analogy with the way classical independence can be defined via classical cumulants. Free cumulants are introduced through the Möbius function. Free product probability spaces are constructed using free cumulants. Marginal and joint tracial convergence of large dimensional random matrices such as the Wigner, elliptic, sample covariance, cross-covariance, Toeplitz, Circulant and Hankel are discussed. Convergence of the empirical spectral distribution is discussed for symmetric matrices. Asymptotic freeness results for random matrices, including some recent ones, are discussed in detail. These clarify the structure of the limits for joint convergence of random matrices. Asymptotic freeness of independent sample covariance matrices is also demonstrated via embedding into Wigner matrices. Exercises, at advanced undergraduate and graduate level, are provided in each chapter.

An Introduction to Random Matrices

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Publisher : Cambridge University Press
ISBN 13 : 0521194520
Total Pages : 507 pages
Book Rating : 4.5/5 (211 download)

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Book Synopsis An Introduction to Random Matrices by : Greg W. Anderson

Download or read book An Introduction to Random Matrices written by Greg W. Anderson and published by Cambridge University Press. This book was released on 2010 with total page 507 pages. Available in PDF, EPUB and Kindle. Book excerpt: A rigorous introduction to the basic theory of random matrices designed for graduate students with a background in probability theory.

Patterned Random Matrices

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Author :
Publisher : CRC Press
ISBN 13 : 0429948891
Total Pages : 269 pages
Book Rating : 4.4/5 (299 download)

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Book Synopsis Patterned Random Matrices by : Arup Bose

Download or read book Patterned Random Matrices written by Arup Bose and published by CRC Press. This book was released on 2018-05-23 with total page 269 pages. Available in PDF, EPUB and Kindle. Book excerpt: Large dimensional random matrices (LDRM) with specific patterns arise in econometrics, computer science, mathematics, physics, and statistics. This book provides an easy initiation to LDRM. Through a unified approach, we investigate the existence and properties of the limiting spectral distribution (LSD) of different patterned random matrices as the dimension grows. The main ingredients are the method of moments and normal approximation with rudimentary combinatorics for support. Some elementary results from matrix theory are also used. By stretching the moment arguments, we also have a brush with the intriguing but difficult concepts of joint convergence of sequences of random matrices and its ramifications. This book covers the Wigner matrix, the sample covariance matrix, the Toeplitz matrix, the Hankel matrix, the sample autocovariance matrix and the k-Circulant matrices. Quick and simple proofs of their LSDs are provided and it is shown how the semi-circle law and the March enko-Pastur law arise as the LSDs of the first two matrices. Extending the basic approach, we also establish interesting limits for some triangular matrices, band matrices, balanced matrices, and the sample autocovariance matrix. We also study the joint convergence of several patterned matrices, and show that independent Wigner matrices converge jointly and are asymptotically free of other patterned matrices. Arup Bose is a Professor at the Indian Statistical Institute, Kolkata, India. He is a distinguished researcher in Mathematical Statistics and has been working in high-dimensional random matrices for the last fifteen years. He has been the Editor of Sankyhā for several years and has been on the editorial board of several other journals. He is a Fellow of the Institute of Mathematical Statistics, USA and all three national science academies of India, as well as the recipient of the S.S. Bhatnagar Award and the C.R. Rao Award. His forthcoming books are the monograph, Large Covariance and Autocovariance Matrices (with Monika Bhattacharjee), to be published by Chapman & Hall/CRC Press, and a graduate text, U-statistics, M-estimates and Resampling (with Snigdhansu Chatterjee), to be published by Hindustan Book Agency.

Introduction to Random Matrices

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Author :
Publisher : Springer
ISBN 13 : 3319708856
Total Pages : 122 pages
Book Rating : 4.3/5 (197 download)

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Book Synopsis Introduction to Random Matrices by : Giacomo Livan

Download or read book Introduction to Random Matrices written by Giacomo Livan and published by Springer. This book was released on 2018-01-16 with total page 122 pages. Available in PDF, EPUB and Kindle. Book excerpt: Modern developments of Random Matrix Theory as well as pedagogical approaches to the standard core of the discipline are surprisingly hard to find in a well-organized, readable and user-friendly fashion. This slim and agile book, written in a pedagogical and hands-on style, without sacrificing formal rigor fills this gap. It brings Ph.D. students in Physics, as well as more senior practitioners, through the standard tools and results on random matrices, with an eye on most recent developments that are not usually covered in introductory texts. The focus is mainly on random matrices with real spectrum.The main guiding threads throughout the book are the Gaussian Ensembles. In particular, Wigner’s semicircle law is derived multiple times to illustrate several techniques (e.g., Coulomb gas approach, replica theory).Most chapters are accompanied by Matlab codes (stored in an online repository) to guide readers through the numerical check of most analytical results.

Patterned Random Matrices

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Author :
Publisher : CRC Press
ISBN 13 : 0429948883
Total Pages : 329 pages
Book Rating : 4.4/5 (299 download)

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Book Synopsis Patterned Random Matrices by : Arup Bose

Download or read book Patterned Random Matrices written by Arup Bose and published by CRC Press. This book was released on 2018-05-23 with total page 329 pages. Available in PDF, EPUB and Kindle. Book excerpt: Large dimensional random matrices (LDRM) with specific patterns arise in econometrics, computer science, mathematics, physics, and statistics. This book provides an easy initiation to LDRM. Through a unified approach, we investigate the existence and properties of the limiting spectral distribution (LSD) of different patterned random matrices as the dimension grows. The main ingredients are the method of moments and normal approximation with rudimentary combinatorics for support. Some elementary results from matrix theory are also used. By stretching the moment arguments, we also have a brush with the intriguing but difficult concepts of joint convergence of sequences of random matrices and its ramifications. This book covers the Wigner matrix, the sample covariance matrix, the Toeplitz matrix, the Hankel matrix, the sample autocovariance matrix and the k-Circulant matrices. Quick and simple proofs of their LSDs are provided and it is shown how the semi-circle law and the March enko-Pastur law arise as the LSDs of the first two matrices. Extending the basic approach, we also establish interesting limits for some triangular matrices, band matrices, balanced matrices, and the sample autocovariance matrix. We also study the joint convergence of several patterned matrices, and show that independent Wigner matrices converge jointly and are asymptotically free of other patterned matrices. Arup Bose is a Professor at the Indian Statistical Institute, Kolkata, India. He is a distinguished researcher in Mathematical Statistics and has been working in high-dimensional random matrices for the last fifteen years. He has been the Editor of Sankyhā for several years and has been on the editorial board of several other journals. He is a Fellow of the Institute of Mathematical Statistics, USA and all three national science academies of India, as well as the recipient of the S.S. Bhatnagar Award and the C.R. Rao Award. His forthcoming books are the monograph, Large Covariance and Autocovariance Matrices (with Monika Bhattacharjee), to be published by Chapman & Hall/CRC Press, and a graduate text, U-statistics, M-estimates and Resampling (with Snigdhansu Chatterjee), to be published by Hindustan Book Agency.

Random Matrices and Non-Commutative Probability

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Author :
Publisher : CRC Press
ISBN 13 : 1000458822
Total Pages : 420 pages
Book Rating : 4.0/5 (4 download)

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Book Synopsis Random Matrices and Non-Commutative Probability by : Arup Bose

Download or read book Random Matrices and Non-Commutative Probability written by Arup Bose and published by CRC Press. This book was released on 2021-10-26 with total page 420 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is an introductory book on Non-Commutative Probability or Free Probability and Large Dimensional Random Matrices. Basic concepts of free probability are introduced by analogy with classical probability in a lucid and quick manner. It then develops the results on the convergence of large dimensional random matrices, with a special focus on the interesting connections to free probability. The book assumes almost no prerequisite for the most part. However, familiarity with the basic convergence concepts in probability and a bit of mathematical maturity will be helpful. Combinatorial properties of non-crossing partitions, including the Möbius function play a central role in introducing free probability. Free independence is defined via free cumulants in analogy with the way classical independence can be defined via classical cumulants. Free cumulants are introduced through the Möbius function. Free product probability spaces are constructed using free cumulants. Marginal and joint tracial convergence of large dimensional random matrices such as the Wigner, elliptic, sample covariance, cross-covariance, Toeplitz, Circulant and Hankel are discussed. Convergence of the empirical spectral distribution is discussed for symmetric matrices. Asymptotic freeness results for random matrices, including some recent ones, are discussed in detail. These clarify the structure of the limits for joint convergence of random matrices. Asymptotic freeness of independent sample covariance matrices is also demonstrated via embedding into Wigner matrices. Exercises, at advanced undergraduate and graduate level, are provided in each chapter.

Random Matrices and the Statistical Theory of Energy Levels

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Publisher : Academic Press
ISBN 13 : 1483258564
Total Pages : 270 pages
Book Rating : 4.4/5 (832 download)

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Book Synopsis Random Matrices and the Statistical Theory of Energy Levels by : M. L. Mehta

Download or read book Random Matrices and the Statistical Theory of Energy Levels written by M. L. Mehta and published by Academic Press. This book was released on 2014-05-12 with total page 270 pages. Available in PDF, EPUB and Kindle. Book excerpt: Random Matrices and the Statistical Theory of Energy Levels focuses on the processes, methodologies, calculations, and approaches involved in random matrices and the statistical theory of energy levels, including ensembles and density and correlation functions. The publication first elaborates on the joint probability density function for the matrix elements and eigenvalues, including the Gaussian unitary, symplectic, and orthogonal ensembles and time-reversal invariance. The text then examines the Gaussian ensembles, as well as the asymptotic formula for the level density and partition function. The manuscript elaborates on the Brownian motion model, circuit ensembles, correlation functions, thermodynamics, and spacing distribution of circular ensembles. Topics include continuum model for the spacing distribution, thermodynamic quantities, joint probability density function for the eigenvalues, stationary and nonstationary ensembles, and ensemble averages. The publication then examines the joint probability density functions for two nearby spacings and invariance hypothesis and matrix element correlations. The text is a valuable source of data for researchers interested in random matrices and the statistical theory of energy levels.

Large Covariance and Autocovariance Matrices

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Publisher : CRC Press
ISBN 13 : 1351398164
Total Pages : 272 pages
Book Rating : 4.3/5 (513 download)

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Book Synopsis Large Covariance and Autocovariance Matrices by : Arup Bose

Download or read book Large Covariance and Autocovariance Matrices written by Arup Bose and published by CRC Press. This book was released on 2018-07-03 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: Large Covariance and Autocovariance Matrices brings together a collection of recent results on sample covariance and autocovariance matrices in high-dimensional models and novel ideas on how to use them for statistical inference in one or more high-dimensional time series models. The prerequisites include knowledge of elementary multivariate analysis, basic time series analysis and basic results in stochastic convergence. Part I is on different methods of estimation of large covariance matrices and auto-covariance matrices and properties of these estimators. Part II covers the relevant material on random matrix theory and non-commutative probability. Part III provides results on limit spectra and asymptotic normality of traces of symmetric matrix polynomial functions of sample auto-covariance matrices in high-dimensional linear time series models. These are used to develop graphical and significance tests for different hypotheses involving one or more independent high-dimensional linear time series. The book should be of interest to people in econometrics and statistics (large covariance matrices and high-dimensional time series), mathematics (random matrices and free probability) and computer science (wireless communication). Parts of it can be used in post-graduate courses on high-dimensional statistical inference, high-dimensional random matrices and high-dimensional time series models. It should be particularly attractive to researchers developing statistical methods in high-dimensional time series models. Arup Bose is a professor at the Indian Statistical Institute, Kolkata, India. He is a distinguished researcher in mathematical statistics and has been working in high-dimensional random matrices for the last fifteen years. He has been editor of Sankhyā for several years and has been on the editorial board of several other journals. He is a Fellow of the Institute of Mathematical Statistics, USA and all three national science academies of India, as well as the recipient of the S.S. Bhatnagar Award and the C.R. Rao Award. His first book Patterned Random Matrices was also published by Chapman & Hall. He has a forthcoming graduate text U-statistics, M-estimates and Resampling (with Snigdhansu Chatterjee) to be published by Hindustan Book Agency. Monika Bhattacharjee is a post-doctoral fellow at the Informatics Institute, University of Florida. After graduating from St. Xavier's College, Kolkata, she obtained her master’s in 2012 and PhD in 2016 from the Indian Statistical Institute. Her thesis in high-dimensional covariance and auto-covariance matrices, written under the supervision of Dr. Bose, has received high acclaim.

Random Matrices

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Author :
Publisher : Elsevier
ISBN 13 : 008047411X
Total Pages : 707 pages
Book Rating : 4.0/5 (84 download)

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Book Synopsis Random Matrices by : Madan Lal Mehta

Download or read book Random Matrices written by Madan Lal Mehta and published by Elsevier. This book was released on 2004-10-06 with total page 707 pages. Available in PDF, EPUB and Kindle. Book excerpt: Random Matrices gives a coherent and detailed description of analytical methods devised to study random matrices. These methods are critical to the understanding of various fields in in mathematics and mathematical physics, such as nuclear excitations, ultrasonic resonances of structural materials, chaotic systems, the zeros of the Riemann and other zeta functions. More generally they apply to the characteristic energies of any sufficiently complicated system and which have found, since the publication of the second edition, many new applications in active research areas such as quantum gravity, traffic and communications networks or stock movement in the financial markets. This revised and enlarged third edition reflects the latest developements in the field and convey a greater experience with results previously formulated. For example, the theory of skew-orthogoanl and bi-orthogonal polynomials, parallel to that of the widely known and used orthogonal polynomials, is explained here for the first time. - Presentation of many new results in one place for the first time - First time coverage of skew-orthogonal and bi-orthogonal polynomials and their use in the evaluation of some multiple integrals - Fredholm determinants and Painlevé equations - The three Gaussian ensembles (unitary, orthogonal, and symplectic); their n-point correlations, spacing probabilities - Fredholm determinants and inverse scattering theory - Probability densities of random determinants

Special Matrices of Mathematical Physics

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Publisher : World Scientific
ISBN 13 : 9789812799838
Total Pages : 344 pages
Book Rating : 4.7/5 (998 download)

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Book Synopsis Special Matrices of Mathematical Physics by : Ruben Aldrovandi

Download or read book Special Matrices of Mathematical Physics written by Ruben Aldrovandi and published by World Scientific. This book was released on 2001 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt: Ch. 1. Some fundamental notions. 1.1. Definitions. 1.2. Components of a matrix. 1.3. Matrix functions. 1.4. Normal matrices -- ch. 2. Evolving systems -- ch. 3. Markov chains. 3.1. Non-negative matrices. 3.2. General properties -- ch. 4. Glass transition -- ch. 5. The Kerner model. 5.1. A simple example: Se-As glass -- ch. 6. Formal developments. 6.1. Spectral aspects. 6.2. Reducibility and regularity. 6.3. Projectors and asymptotics. 6.4. Continuum time -- ch. 7. Equilibrium, dissipation and ergodicity. 7.1. Recurrence, transience and periodicity. 7.2. Detailed balancing and reversibility. 7.3. Ergodicity -- ch. 8. Prelude -- ch. 9. Definition and main properties. 9.1. Bases. 9.2. Double Fourier transform. 9.3. Random walks -- ch. 10. Discrete quantum mechanics. 10.1. Introduction. 10.2. Weyl-Heisenberg groups. 10.3. Weyl-Wigner transformations. 10.4. Braiding and quantum groups -- ch. 11. Quantum symplectic structure. 11.1. Matrix differential geometry. 11.2. The symplectic form. 11.3. The quantum fabric -- ch. 12. An organizing tool -- ch. 13. Bell polynomials. 13.1. Definition and elementary properties. 13.2. The matrix representation. 13.3. The Lagrange inversion formula. 13.4. Developments -- ch. 14. Determinants and traces. 14.1. Introduction. 14.2. Symmetric functions. 14.3. Polynomials. 14.4. Characteristic polynomials. 14.5. Lie algebras invariants -- ch. 15. Projectors and iterates. 15.1. Projectors, revisited. 15.2. Continuous iterates -- ch. 16. Gases: real and ideal. 16.1. Microcanonical ensemble. 16.2. The canonical ensemble. 16.3. The grand canonical ensemble. 16.4. Braid statistics. 16.5. Condensation theories. 16.6. The Fredholm formalism.

Large Random Matrices: Lectures on Macroscopic Asymptotics

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Author :
Publisher : Springer
ISBN 13 : 3540698973
Total Pages : 296 pages
Book Rating : 4.5/5 (46 download)

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Book Synopsis Large Random Matrices: Lectures on Macroscopic Asymptotics by : Alice Guionnet

Download or read book Large Random Matrices: Lectures on Macroscopic Asymptotics written by Alice Guionnet and published by Springer. This book was released on 2009-04-20 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: Random matrix theory has developed in the last few years, in connection with various fields of mathematics and physics. These notes emphasize the relation with the problem of enumerating complicated graphs, and the related large deviations questions. Such questions are also closely related with the asymptotic distribution of matrices, which is naturally defined in the context of free probability and operator algebra. The material of this volume is based on a series of nine lectures given at the Saint-Flour Probability Summer School 2006. Lectures were also given by Maury Bramson and Steffen Lauritzen.

Random Matrix Theory, Interacting Particle Systems and Integrable Systems

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Publisher : Cambridge University Press
ISBN 13 : 1107079926
Total Pages : 539 pages
Book Rating : 4.1/5 (7 download)

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Book Synopsis Random Matrix Theory, Interacting Particle Systems and Integrable Systems by : Percy Deift

Download or read book Random Matrix Theory, Interacting Particle Systems and Integrable Systems written by Percy Deift and published by Cambridge University Press. This book was released on 2014-12-15 with total page 539 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume includes review articles and research contributions on long-standing questions on universalities of Wigner matrices and beta-ensembles.

A First Course in Random Matrix Theory

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Publisher : Cambridge University Press
ISBN 13 : 1108488080
Total Pages : 371 pages
Book Rating : 4.1/5 (84 download)

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Book Synopsis A First Course in Random Matrix Theory by : Marc Potters

Download or read book A First Course in Random Matrix Theory written by Marc Potters and published by Cambridge University Press. This book was released on 2020-12-03 with total page 371 pages. Available in PDF, EPUB and Kindle. Book excerpt: An intuitive, up-to-date introduction to random matrix theory and free calculus, with real world illustrations and Big Data applications.

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Author :
Publisher : World Scientific
ISBN 13 :
Total Pages : 1131 pages
Book Rating : 4./5 ( download)

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Book Synopsis by :

Download or read book written by and published by World Scientific. This book was released on with total page 1131 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Circulant Matrices

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Author :
Publisher : John Wiley & Sons
ISBN 13 :
Total Pages : 276 pages
Book Rating : 4.3/5 (91 download)

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Book Synopsis Circulant Matrices by : Philip J. Davis

Download or read book Circulant Matrices written by Philip J. Davis and published by John Wiley & Sons. This book was released on 1979 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: