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Matrix Mathematics: Theory, Facts, and Formulas - Second Edition
by Dennis S. BernsteinWhen first published in 2005, Matrix Mathematics quickly became the essential reference book for users of matrices in all branches of engineering, science, and applied mathematics. In this fully updated and expanded edition, the author brings together the latest results on matrix theory to make this the most complete, current, and easy-to-use book on matrices. Each chapter describes relevant background theory followed by specialized results. Hundreds of identities, inequalities, and matrix facts are stated clearly and rigorously with cross references, citations to the literature, and illuminating remarks. Beginning with preliminaries on sets, functions, and relations,Matrix Mathematics covers all of the major topics in matrix theory, including matrix transformations; polynomial matrices; matrix decompositions; generalized inverses; Kronecker and Schur algebra; positive-semidefinite matrices; vector and matrix norms; the matrix exponential and stability theory; and linear systems and control theory. Also included are a detailed list of symbols, a summary of notation and conventions, an extensive bibliography and author index with page references, and an exhaustive subject index. This significantly expanded edition of Matrix Mathematics features a wealth of new material on graphs, scalar identities and inequalities, alternative partial orderings, matrix pencils, finite groups, zeros of multivariable transfer functions, roots of polynomials, convex functions, and matrix norms. Covers hundreds of important and useful results on matrix theory, many never before available in any book Provides a list of symbols and a summary of conventions for easy use Includes an extensive collection of scalar identities and inequalities Features a detailed bibliography and author index with page references Includes an exhaustive subject index with cross-referencing
Matrix Operations for Engineers and Scientists
by Alan JeffreyEngineers and scientists need to have an introduction to the basics of linear algebra in a context they understand. Computer algebra systems make the manipulation of matrices and the determination of their properties a simple matter, and in practical applications such software is often essential. However, using this tool when learning about matrices, without first gaining a proper understanding of the underlying theory, limits the ability to use matrices and to apply them to new problems. This book explains matrices in the detail required by engineering or science students, and it discusses linear systems of ordinary differential equations. These students require a straightforward introduction to linear algebra illustrated by applications to which they can relate. It caters of the needs of undergraduate engineers in all disciplines, and provides considerable detail where it is likely to be helpful. According to the author the best way to understand the theory of matrices is by working simple exercises designed to emphasize the theory, that at the same time avoid distractions caused by unnecessary numerical calculations. Hence, examples and exercises in this book have been constructed in such a way that wherever calculations are necessary they are straightforward. For example, when a characteristic equation occurs, its roots (the eigenvalues of a matrix) can be found by inspection. The author of this book is Alan Jeffrey, Emeritus Professor of mathematics at the University of Newcastle upon Tyne. He has given courses on engineering mathematics at UK and US Universities.
The Matrix Perturbation Method in Quantum Mechanics
by Francisco Soto-Eguibar Braulio Misael Villegas-Martínez Héctor Manuel Moya-CessaThis book provides an alternative approach to time-independent perturbation theory in non-relativistic quantum mechanics. It allows easy application to any initial condition because it is based on an approximation to the evolution operator and may also be used on unitary evolution operators for the unperturbed Hamiltonian in the case where the eigenvalues cannot be found. This flexibility sets it apart from conventional perturbation theory. The matrix perturbation method also gives new theoretical insights; for example, it provides corrections to the energy and wave function in one operation. Another notable highlight is the facility to readily derive a general expression for the normalization constant at m-th order, a significant difference between the approach within and those already in the literature. Another unique aspect of the matrix perturbation method is that it can be extended directly to the Lindblad master equation. The first and second-order corrections are obtained for this equation and the method is generalized for higher orders. An alternative form of the Dyson series, in matrix form instead of integral form, is also obtained. Throughout the book, several benchmark examples and practical applications underscore the potential, accuracy and good performance of this novel approach. Moreover, the method's applicability extends to some specific time-dependent Hamiltonians. This book represents a valuable addition to the literature on perturbation theory in quantum mechanics and is accessible to students and researchers alike.
Matrix Positivity (Cambridge Tracts in Mathematics #221)
by Charles R. Johnson Ronald L. Smith Michael J. TsatsomerosMatrix positivity is a central topic in matrix theory: properties that generalize the notion of positivity to matrices arose from a large variety of applications, and many have also taken on notable theoretical significance, either because they are natural or unifying. This is the first book to provide a comprehensive and up-to-date reference of important material on matrix positivity classes, their properties, and their relations. The matrix classes emphasized in this book include the classes of semipositive matrices, P-matrices, inverse M-matrices, and copositive matrices. This self-contained reference will be useful to a large variety of mathematicians, engineers, and social scientists, as well as graduate students. The generalizations of positivity and the connections observed provide a unique perspective, along with theoretical insight into applications and future challenges. Direct applications can be found in data analysis, differential equations, mathematical programming, computational complexity, models of the economy, population biology, dynamical systems and control theory.
Matrix Theory (Dover Books on Mathematics)
by Joel N. FranklinNot only is matrix theory significant in a wide range of fields mathematical economics, quantum physics, geophysics, electrical network synthesis, crystallography, and structural engineering, among others-but with the vast proliferation of digital computers, knowledge of matrix theory is a must for every modern engineer, mathematician, and scientist. Matrices represent linear transformations from a finiteset of numbers to another finite set of numbers.Since many important problems are linear, and since digital computers with finite memory manipulate only finite sets of numbers, the solution of linear problems by digital computers usually involves matrices. Developed from the author's course on matrix theory at the CaliforniaInstitute of Technology, the book begins with a concise presentation of the theory of determinants, continues with a discussion of classical linear algebra, and an optional chapter on the use of matrices to solve systems of linear triangularizations of Hermitian and nonHermitian matrices, as well as a chapter presenting a proof of the difficult and important matrix theory of Jordan. The book concludes with discussions of variational principles and perturbation theory of matrices, matrix numerical analysis, and an introduction to the subject of linear computations.The book is designed to meet many different needs, and because it is mathematically rigorous, it may be used by students of pure and applied mathematics. Since it is oriented towards applications, it is valuable to students of engineering, science, and the social sciences. And because it contains the basic preparation in matrix theory required for numerical analysis, it can be used by students whose main interest is computers. The book assumes very little mathematical preparation, and except for the single section on the continuous dependence of eigenvalues on matrices, a knowledge of elementary algebra and calculus is sufficient.
Matrix Theory: From Generalized Inverses to Jordan Form (Chapman & Hall/CRC Pure and Applied Mathematics)
by Robert Piziak P.L. OdellIn 1990, the National Science Foundation recommended that every college mathematics curriculum should include a second course in linear algebra. In answer to this recommendation, Matrix Theory: From Generalized Inverses to Jordan Form provides the material for a second semester of linear algebra that probes introductory linear algebra concepts whil
Matrix Theory and Applications for Scientists and Engineers (Dover Books on Mathematics)
by Alexander GrahamA comprehensive text on matrix theory and its applications, this book is intended for a broad range of students in mathematics, engineering, and other areas of science at the university level. Author Alexander Graham avoids a simple catalogue of techniques by exploring the concepts' underlying principles as well as their numerous applications. Many problems elucidate the text, which includes a substantial answer section at the end. The treatment explores matrices, vector spaces, linear transformations, and the rank and determinant of a matrix. Additional topics include linear equations, eigenvectors and eigenvalues, canonical forms and matrix functions, and inverting a matrix. A Solution to Problems Section, References and a Bibliography conclude the treatment.
Matrix Theory and Applications with MATLAB
by Darald J. HartfielDesigned for use in a second course on linear algebra, Matrix Theory and Applications with MATLAB covers the basics of the subject-from a review of matrix algebra through vector spaces to matrix calculus and unitary similarity-in a presentation that stresses insight, understanding, and applications. Among its most outstanding features is the integration of MATLAB throughout the text. Each chapter includes a MATLAB subsection that discusses the various commands used to do the computations in that section and offers code for the graphics and some algorithms used in the text.All of the material is presented from a matrix point of view with enough rigor for students to learn to compose arguments and proofs and adjust the material to cover other problems. The treatment includes optional subsections covering applications, and the final chapters move beyond basic matrix theory to discuss more advanced topics, such as decompositions, positive definite matrices, graphics, and topology.Filled with illustrations, examples, and exercises that reinforce understanding, Matrix Theory and Applications with MATLAB allows readers to experiment and visualize results in a way that no other text does. Its rigor, use of MATLAB, and focus on applications better prepares them to use the material in their future work and research, to extend the material, and perhaps obtain new results of their own.
Matrix Transforms for Computer Games and Animation
by John VinceMatrix transforms are ubiquitous within the world of computer graphics, where they have become an invaluable tool in a programmer's toolkit for solving everything from 2D image scaling to 3D rotation about an arbitrary axis. Virtually every software system and hardware graphics processor uses matrices to undertake operations such as scaling, translation, reflection and rotation. Nevertheless, for some newcomers to the world of computer games and animation, matrix notation can appear obscure and challenging. Matrices and determinants were originally used to solve groups of simultaneous linear equations, and were subsequently embraced by the computer graphics community to describe the geometric operations for manipulating two- and three-dimensional structures. Consequently, to place matrix notation within an historical context, the author provides readers with some useful background to their development, alongside determinants. Although it is assumed that the reader is familiar with everyday algebra and the solution of simultaneous linear equations, Matrix Transforms for Computer Games and Animation does not expect any prior knowledge of matrix notation. It includes chapters on matrix notation, determinants, matrices, 2D transforms, 3D transforms and quaternions, and includes many worked examples to illustrate their practical use.
Matrix Variate Distributions (Monographs and Surveys in Pure and Applied Mathematics #104)
by A K Gupta D K NagarUseful in physics, economics, psychology, and other fields, random matrices play an important role in the study of multivariate statistical methods. Until now, however, most of the material on random matrices could only be found scattered in various statistical journals. Matrix Variate Distributions gathers and systematically presents most of the recent developments in continuous matrix variate distribution theory and includes new results.After a review of the essential background material, the authors investigate the range of matrix variate distributions, including:matrix variate normal distributionWishart distributionMatrix variate t-distributionMatrix variate beta distributionF-distributionMatrix variate Dirichlet distributionMatrix quadratic formsWith its inclusion of new results, Matrix Variate Distributions promises to stimulate further research and help advance the field of multivariate statistical analysis.
Matrix Vector Analysis (Dover Books on Mathematics)
by Richard L. EisenmanThis outstanding text and reference applies matrix ideas to vector methods, using physical ideas to illustrate and motivate mathematical concepts but employing a mathematical continuity of development rather than a physical approach. The author, who taught at the U.S. Air Force Academy, dispenses with the artificial barrier between vectors and matrices--and more generally, between pure and applied mathematics.Motivated examples introduce each idea, with interpretations of physical, algebraic, and geometric contexts, in addition to generalizations to theorems that reflect the essential structure. A combination of matrix and vector methods reinforces both techniques. "Applied" students learn that general theory is a natural and useful culmination of their computations, and "pure" students learn that concrete problems from the physical world have traditionally keynoted abstract intellectual pursuits.Geared toward upper-level undergraduates, the text features approximately 50 provocative problems at each chapter's end that test students' choice of techniques. Each chapter is also followed by about 25 mental exercises that stimulate imaginative reflection. Answers are given to selected questions.
Matroids: A Geometric Introduction
by Gary Gordon Jennifer McnultyMatroid theory is a vibrant area of research that provides a unified way to understand graph theory, linear algebra and combinatorics via finite geometry. This book provides the first comprehensive introduction to the field which will appeal to undergraduate students and to any mathematician interested in the geometric approach to matroids. Written in a friendly, fun-to-read style and developed from the authors' own undergraduate courses, the book is ideal for students. Beginning with a basic introduction to matroids, the book quickly familiarizes the reader with the breadth of the subject, and specific examples are used to illustrate the theory and to help students see matroids as more than just generalizations of graphs. Over 300 exercises are included, with many hints and solutions so students can test their understanding of the materials covered. The authors have also included several projects and open-ended research problems for independent study.
A Matter of Life and Death: Hunting in Contemporary Vermont
by Marc BoglioliIn A Matter of Life and Death, anthropologist Marc Boglioli puts a human face on a group widely regarded as morally suspect, one that currently stands in the crossfire of America's so-called culture wars. Drawing on extensive ethnographic fieldwork in Addison County, Vermont, which took him from hunting camps and sporting goods stores to local bars and kitchen tables, Boglioli focuses on how contemporary hunters, women as well as men, understand their relationship to their prey. He shows how hunters' attitudes toward animals flow directly from the rural lifeways they have continued to maintain in the face of encroaching urban sensibilities. The result is a rare glimpse into a culture that experiences wild animals in a way that is at once violent, consumptive, and respectful, and that regards hunting as an enduring link to a vanishing past. It is a book that will challenge readers―hunters, non-hunters, and anti-hunters alike―to reconsider what constitutes a morally appropriate relationship with the non-human residents of this planet.
Mauritius Mathematics Grade 7 (Part-I) - MIE
by Mauritius Institute of EducationMauritius Mathematics Grade 7 (Part-I) Textbook Mauritius Institute of Education.
Mauritius Mathematics Grade 7 (Part-II) - MIE
by Mauritius Institute of EducationThe Mathematics Grade 7 textbook by Mauritius Institute of Education.
Max-linear Systems: Theory and Algorithms
by Peter ButkovičRecent years have seen a significant rise of interest in max-linear theory and techniques. Specialised international conferences and seminars or special sessions devoted to max-algebra have been organised. This book aims to provide a first detailed and self-contained account of linear-algebraic aspects of max-algebra for general (that is both irreducible and reducible) matrices. Among the main features of the book is the presentation of the fundamental max-algebraic theory (Chapters 1-4), often scattered in research articles, reports and theses, in one place in a comprehensive and unified form. This presentation is made with all proofs and in full generality (that is for both irreducible and reducible matrices). Another feature is the presence of advanced material (Chapters 5-10), most of which has not appeared in a book before and in many cases has not been published at all. Intended for a wide-ranging readership, this book will be useful for anyone with basic mathematical knowledge (including undergraduate students) who wish to learn fundamental max-algebraic ideas and techniques. It will also be useful for researchers working in tropical geometry or idempotent analysis.
Max Planck und die moderne Physik
by Dieter HoffmannMax Planck (1858-1947) zählt zu den herausragenden Physikern des zwanzigsten Jahrhunderts. Seine Quantenhypothese wurde zur Grundlage für die Entwicklung der Quantentheorie und bildet eine der zentralen Säulen der modernen Physik. Damit ist Plancks Wirken jedoch keineswegs erschöpfend charakterisiert: In diesem Buch arbeiten anerkannte Physikhistoriker das gesamte Spektrum von Plancks physikalischem Schaffen heraus und würdigen seine Bedeutung für die Entwicklung der modernen Physik.
Max Weber: Collected Methodological Writings (Weber in Translation)
by Sam Whimster Hans Henrik BruunWeber’s methodological writings form the bedrock of key ideas across the social sciences. His discussion of value freedom and value commitment, causality, understanding and explanation, theory building and ideal types have been of fundamental importance, and their impact remains undiminished today. These ideas influence the current research practice of sociologists, historians, economists and political scientists and are central to debates in the philosophy of social science. But, until now, Weber's extensive writings on methodology have lacked a comprehensive publication. Edited by two of the world's leading Weber scholars, Collected Methodological Writings will provide a completely new, accurate and reliable translation of Weber’s extensive output, including previously untranslated letters. Accompanying editorial commentary explains the context of, and interconnections between, all these writings, and additional useful features include a glossary of German terms and an English key, endnotes, bibliography, and person and subject indexes.
Maximal Solvable Subgroups of Finite Classical Groups (Lecture Notes in Mathematics #2346)
by Mikko KorhonenThis book studies maximal solvable subgroups of classical groups over finite fields. It provides the first modern account of Camille Jordan's classical results, and extends them, giving a classification of maximal irreducible solvable subgroups of general linear groups, symplectic groups, and orthogonal groups over arbitrary finite fields. A subgroup of a group G is said to be maximal solvable if it is maximal among the solvable subgroups of G. The history of this notion goes back to Jordan’s Traité (1870), in which he provided a classification of maximal solvable subgroups of symmetric groups. The main difficulty is in the primitive case, which leads to the problem of classifying maximal irreducible solvable subgroups of general linear groups over a field of prime order. One purpose of this monograph is expository: to give a proof of Jordan’s classification in modern terms. More generally, the aim is to generalize these results to classical groups over arbitrary finite fields, and to provide other results of interest related to irreducible solvable matrix groups. The text will be accessible to graduate students and researchers interested in primitive permutation groups, irreducible matrix groups, and related topics in group theory and representation theory. The detailed introduction will appeal to those interested in the historical background of Jordan’s work.
The Maximal Subgroups of the Low-Dimensional Finite Classical Groups
by John N. Bray Derek F. Holt Colva M. Roney-DougalThis book classifies the maximal subgroups of the almost simple finite classical groups in dimension up to 12; it also describes the maximal subgroups of the almost simple finite exceptional groups with socle one of Sz(q), G2(q), 2G2(q) or 3D4(q). Theoretical and computational tools are used throughout, with downloadable Magma code provided. The exposition contains a wealth of information on the structure and action of the geometric subgroups of classical groups, but the reader will also encounter methods for analysing the structure and maximality of almost simple subgroups of almost simple groups. Additionally, this book contains detailed information on using Magma to calculate with representations over number fields and finite fields. Featured within are previously unseen results and over 80 tables describing the maximal subgroups, making this volume an essential reference for researchers. It also functions as a graduate-level textbook on finite simple groups, computational group theory and representation theory.
Maximum Likelihood Estimation and Inference
by Russell B. MillarThis book takes a fresh look at the popular and well-established method of maximum likelihood for statistical estimation and inference. It begins with an intuitive introduction to the concepts and background of likelihood, and moves through to the latest developments in maximum likelihood methodology, including general latent variable models and new material for the practical implementation of integrated likelihood using the free ADMB software. Fundamental issues of statistical inference are also examined, with a presentation of some of the philosophical debates underlying the choice of statistical paradigm.Key features:Provides an accessible introduction to pragmatic maximum likelihood modelling.Covers more advanced topics, including general forms of latent variable models (including non-linear and non-normal mixed-effects and state-space models) and the use of maximum likelihood variants, such as estimating equations, conditional likelihood, restricted likelihood and integrated likelihood.Adopts a practical approach, with a focus on providing the relevant tools required by researchers and practitioners who collect and analyze real data.Presents numerous examples and case studies across a wide range of applications including medicine, biology and ecology.Features applications from a range of disciplines, with implementation in R, SAS and/or ADMB.Provides all program code and software extensions on a supporting website.Confines supporting theory to the final chapters to maintain a readable and pragmatic focus of the preceding chapters. This book is not just an accessible and practical text about maximum likelihood, it is a comprehensive guide to modern maximum likelihood estimation and inference. It will be of interest to readers of all levels, from novice to expert. It will be of great benefit to researchers, and to students of statistics from senior undergraduate to graduate level. For use as a course text, exercises are provided at the end of each chapter.
Maximum Likelihood Estimation for Sample Surveys (Chapman & Hall/CRC Monographs on Statistics and Applied Probability)
by Raymond L. Chambers David G. Steel Suojin Wang Alan WelshSample surveys provide data used by researchers in a large range of disciplines to analyze important relationships using well-established and widely used likelihood methods. The methods used to select samples often result in the sample differing in important ways from the target population and standard application of likelihood methods can lead to
Maximum Principles and Geometric Applications
by Luis J. Alías Paolo Mastrolia Marco RigoliThis monograph presents an introduction to some geometric and analytic aspects of the maximum principle. In doing so, it analyses with great detail the mathematical tools and geometric foundations needed to develop the various new forms that are presented in the first chapters of the book. In particular, a generalization of the Omori-Yau maximum principle to a wide class of differential operators is given, as well as a corresponding weak maximum principle and its equivalent open form and parabolicity as a special stronger formulation of the latter. In the second part, the attention focuses on a wide range of applications, mainly to geometric problems, but also on some analytic (especially PDEs) questions including: the geometry of submanifolds, hypersurfaces in Riemannian and Lorentzian targets, Ricci solitons, Liouville theorems, uniqueness of solutions of Lichnerowicz-type PDEs and so on. Maximum Principles and Geometric Applications is written in an easy style making it accessible to beginners. The reader is guided with a detailed presentation of some topics of Riemannian geometry that are usually not covered in textbooks. Furthermore, many of the results and even proofs of known results are new and lead to the frontiers of a contemporary and active field of research.
Maxwell's Demon 2 Entropy, Classical and Quantum Information, Computing
by Harvey S Leff Andrew F RexOver 130 years ago, James Clerk Maxwell introduced his hypothetical "demon" as a challenge to the scope of the second law of thermodynamics. Fascination with the demon persisted throughout the development of statistical and quantum physics, information theory, and computer science, and links have been established between Maxwell's demon and each of
May Contain Lies: How Stories, Statistics, and Studies Exploit Our Biases—And What We Can Do about It
by Alex EdmansHow our biases cause us to fall for misinformation—and how to combat it. Our lives are minefields of misinformation. It ripples through our social media feeds, our daily headlines, and the pronouncements of politicians, executives, and authors. Stories, statistics, and studies are everywhere, allowing people to find evidence to support whatever position they want. Many of these sources are flawed, yet by playing on our emotions and preying on our biases, they can gain widespread acceptance, warp our views, and distort our decisions. In this eye-opening book, renowned economist Alex Edmans teaches us how to separate fact from fiction. Using colorful examples—from a wellness guru’s tragic but fabricated backstory to the blunders that led to the Deepwater Horizon disaster to the diet that ensnared millions yet hastened its founder’s death—Edmans highlights the biases that cause us to mistake statements for facts, facts for data, data for evidence, and evidence for proof. Armed with the knowledge of what to guard against, he then provides a practical guide to combat this tide of misinformation. Going beyond simply checking the facts and explaining individual statistics, Edmans explores the relationships between statistics—the science of cause and effect—ultimately training us to think smarter, sharper, and more critically. May Contain Lies is an essential read for anyone who wants to make better sense of the world and better decisions.