Book Description

"Metric geometry" is an approach to geometry based on the notionof length on a topological space. This approach experienced avery fast development in the last few decades and penetratedinto many other mathematical disciplines, such as group theory,dynamical systems, and partial differential equations.

The objective of this graduate textbook is twofold: to give adetailed exposition of basic notions and techniques used in thetheory of length spaces, and, more generally, to offer anelementary introduction into a broad variety of geometricaltopics related to the notion of distance, including Riemannianand Carnot-Caratheodory metrics, the hyperbolic plane,distance-volume inequalities, asymptotic geometry (large scale,coarse), Gromov hyperbolic spaces, convergence of metric spaces,and Alexandrov spaces (non-positively and non-negatively curvedspaces). The authors tend to work with "easy-to-touch"mathematical objects using "easy-to-visualize" methods.

The authors set a challenging goal of making the core parts ofthe book accessible to first-year graduate students. Most newconcepts and methods are introduced and illustrated usingsimplest cases and avoiding technicalities. The book containsmany exercises, which form a vital part of exposition. ... Read more

Reviews (1)

Interesting, but a bit uneven
The authors present a self-contained treatment of the geometry of length spaces. They begin with the definition of length spaces, and by the final chapter cover almost all of the material in the well known survey paper by Burago, Gromov, and Perel'man. Overall, the book provides an interesting and and accessible introduction to an important class of spaces (length spaces arise as limits of sequences of Riemannian manifolds). I found it exceedingly interesting that one can build analogues of most of the concepts from Riemannian geometry using only the intrinsic metric. There is great emphasis on exposition in this book; Burago makes great effort to motivate definitions and provide interesting examples. The reader is cautioned, however, that numerous typographic errors are to be found (some in fundamental definitions).

My only complaint is the uneveness of the treatment. The authors spend two chapters developing some of the basic tools of Riemannian geometry in the setting of surfaces in R^3. It seems unlikely to me that a reader who is interested in length spaces would not already have at least passing aquaintence with Riemannian geometry. This, however, is a minor point, and I can recommend the book very highly. ... Read more

Book Description

Unified treatment of general theory of generalized inversion, including topics ranging from the most traditional to the most contemporary. Emphasis is placed on the utility of the concept of generalized inversion by presenting many diverse applications in which generalized inversion plays an integral role. Exercises. Bibliography. 19 illustrations.
... Read more

Book Description

A complete, self-contained introduction to matrix analysis theory and practice

Matrix methods have evolved from a tool for expressing statistical problems to an indispensable part of the development, understanding, and use of various types of complex statistical analyses. As such, they have become a vital part of any statistical education. Unfortunately, matrix methods are usually treated piecemeal in courses on everything from regression analysis to stochastic processes. Matrix Analysis for Statistics offers a unique view of matrix analysis theory and methods as a whole.

Professor James R. Schott provides in-depth, step-by-step coverage of the most common matrix methods now used in statistical applications, including eigenvalues and eigenvectors, the Moore-Penrose inverse, matrix differentiation, the distribution of quadratic forms, and more. The subject matter is presented in a theorem/proof format, and every effort has been made to ease the transition from one topic to another. Proofs are easy to follow, and the author carefully justifies every step. Accessible even for readers with a cursory background in statistics, the text uses examples that are familiar and easy to understand. Other key features that make this the ideal introduction to matrix analysis theory and practice include:

• Self-contained chapters for flexibility in topic choice.
• Extensive examples and chapter-end practice exercises.
• Optional sections for mathematically advanced readers.

Reviews (1)

Use symbols in a confusing way
I can't recommend this book.

One problem is that the author uses symbol in a confusing way: declare without define, define without explaination, especially in the examples related to linear regression. It seems he just pick up symbols at random whenever he needs one in the middle of a proof, make an equation out of it. It's your role to figure out from the equation what this symbol is and why the author introduce this symbol.
More than that symbols are not used consistently. Z1, for example, is used to represent column vectors, row vectors, orthornormal vectors and centeralized observation vectors in very close sections without explanation. A vector just defined as a column vector can be multiplied with another column vector. Seems the author doesn't pay much attentions to such "small" aspects and this makes the book really hard to read.

Another problem is you won't know what the author is trying to do only after the whole algebraic series is finished.The objective is always the last to come. Won't it be nice to just add a few words before each a few steps to say what you are planning to do?

The last complaint is that there's not a simple geometric graph in this book. How can you imagine a book on the subject of both statistics and matrix don't have a illustrative graph on it? ... Read more

Book Description

This book is a revised version of the first edition, originally published by Prentice Hall in 1962 and regarded as a classic in its field. In some places, newer research results, e.g. results on weak regular splittings, have been incorported in the revision, and in other places, new material has been added in the chapters, as well as at the end of chapters, in the form of additional up-to-date references and some recent theorems to give the reader some newer directions to pursue. The material in the new chapters is basically self-contained and more exercises have been provided for the readers. While the original version was more linear algebra oriented, the revision attempts to emphasize tools from other areas, such as approximation theory and conformal mapping theory, to access newer results of interest. The book should be of great interest to researchers and graduate students in the field of numerical analysis. ... Read more

Reviews (1)

Excellent, but very dated
The Editorial Review on Amazon's site says: "While the original version was more linear algebra oriented, the revision attempts to emphasize tools from other areas, such as approximation theory and conformal mapping theory, to access newer results of interest". These remarks give an incorrect flavor for the book. Of approximately 350 references, less than a dozen are post 1990. The vast majority are from 1950 to 1965. The author repeatedly refers to "more recent" results with references to the early 1960's. So much for "newer results". The author's remarks in the preface to the new edition are much more informative: "...just what could easily be added [to the new edition]. For example, even a modest treatment of finite elements...was questionable. This was also the case for multigrid methods, Krylov subspace methods, preconditioning methods, and incomplete factorization methods. In the end, only a few items were added... These items include ovals of Cassini, a semi-iterative analysis of SOR methods,.... and matrix rational approximations to exp(-z)."

You will notice that "conformal mapping" didn't make the grade as a centerpiece of the new edition, and neither is it mentioned in the Table of Contents, nor the Index.

For what is here, the exposition is clear and helpful. However, the reader won't gain much perspective on new developments. ... Read more

Book Description

This volume provides a down-to-earth, easily understandable guide to techniques of matrix theory, which are widely used throughout engineering and the physical, life, and social sciences.Fully up-to-date, the book covers a wide range of topics, from basic matrix algebra to such advanced concepts as generalized inverses and Hadamard matrices, and applications to error-correcting codes, control theory, and linear programming.Results are illustrated with many examples drawn from diverse areas of application. Numerous exercisesare included to clarify the material presented in the text, which is suitable for undergraduates and graduates alike. Researchers will also benefit from the accessible accounts of advanced matrix techniques. ... Read more

Book Description

Matrices are used in many areas such as staustics, econometrics, mathematics, natural sciences and engineering. This handbook provides a collection of numerous results for easy reference in one source, together with a comprehensive dictionary of matrices and related terms. The book is structured and cross-referenced so that specific results and information are easy to locate References for proofs and computational algorithms are provided. Researchers, professionals and students using matrix techniques will find this book to be an invaluable resource for their work. Contents

• Definitions, Notation, Terminology
• Rules for Matrix Operations
• Matrix Valued Functions of a Matrix
• Trace, Determinant and Rank of a Matrix
• Eigenvalues and Singular Values
• Matrix Decompositions and Canonical Forms
• Vectorization Operators
• Vector and Matrix Norms
• Properties of Special Matrices
• Vector and Matrix Derivatives
• Polynomials, Power Series and Matrices

Reviews (1)

An extensive collection of results and definitions
This book is in fact an extensive collection of results and definitions on matrices. If you want a book that develops the theory of matrices, this is not what you are looking for. But, if you work with matrices and you need a quick reference to help your research for example, this is the book for you. It is well organised and contains quite a number of results on matrices. At the same time it provides a number of references for additional information on a particular subject. ... Read more

Book Description

Coverage of matrix algebra for economists and students of economics

Matrix Algebra for Applied Economics explains the important tool of matrix algebra for students of economics and practicing economists. It includes examples that demonstrate the foundation operations of matrix algebra and illustrations of using the algebra for a variety of economic problems.

The authors present the scope and basic definitions of matrices, their arithmetic and simple operations, and describe special matrices and their properties, including the analog of division. They provide in-depth coverage of necessary theory and deal with concepts and operations for using matrices in real-life situations. They discuss linear dependence and independence, as well as rank, canonical forms, generalized inverses, eigenroots, and vectors. Topics of prime interest to economists are shown to be simplified using matrix algebra in linear equations, regression, linear models, linear programming, and Markov chains.

Highlights include:

• Numerous examples of real-world applications
• Challenging exercises throughout the book
• Mathematics understandable to readers of all backgrounds
• Extensive up-to-date reference material

Matrix Algebra for Applied Economics provides excellent guidance for advanced undergraduate students and also graduate students. Practicing economists who want to sharpen their skills will find this book both practical and easy-to-read, no matter what their applied interests. ... Read more

Reviews (1)

Very easy to read!
This bood can be the introduction level bood for undergraduate students. Authors explain everything very clearly from first chapters. If you already have background about linear algebra this book could be the supplementry material. If you want to focus on the advance econometrics this book would be the good choice for reinforce your matrix ability. ... Read more

Book Description

Provides the user with a step-by-step introduction to Fortran 77, BLAS, LINPACK, and MATLAB. It is a reference that spans several levels of practical matrix computations with a strong emphasis on examples and "hands on" experience. ... Read more

Book Description

Useful 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 distribution· Wishart distribution· Matrix variate t-distribution· Matrix variate beta distribution· F-distribution· Matrix variate Dirichlet distribution· Matrix quadratic formsWith its inclusion of new results, Matrix Variate Distributions promises to stimulate further research and help advance the field of multivariate statistical analysis. ... Read more

Book Description

This new edition of Matrix Methods emphasizes applications to Jordan-canonical forms, differential equations, and least squares. The revision now includes an entire new chapter on inner products, additional material on elementary row applications, and hundreds of new exercises.

* Provides an introduction to the functional approach to programming
* Emphasizes the problem to be solved, not the programming language
* Takes the view that all computer programs are a definition of a function
* Includes exercises for each chapter
* Requires at least a high school algebra level of mathematical sophistication
* A self-contained work
* Can be used as a pre-programming language introduction to the mathematics of computing ... Read more

Reviews (1)

Matrices revealed
This is the best book on matrix algebra I ever read. It so thoroughly unraveled the mysteries of inverting and transposing matrices that I saw the world in a whole new light. I look forward to the movie!! ... Read more

Book Description

Algebraic Riccati equations play a central role in modern control theory and signal processing, essential topics for industrial scientists, engineers, and mathematicians. This book provides a thorough treatment of the theory behind these equations, the solutions arising from continuous and discrete systems, and includes a collection of eight problem areas or applications in which they play a crucial role. ... Read more

Book Description

This book presents the basic ideas of matrix and linear algebra in such a way that users from diverse backgrounds (who have had some exposure to calculus) will understand, by utilizing both algebraic and geometric reasoning. A spiral approach gradually introduces the abstract foundations of the topics involved—linear combination, closure, subspaces, linear independence/dependence, and bases. Opportunities for a variety of applications, and the optional use of MATLAB, provide hands-on explorations of computations and concepts.Chapter topics include matrices, linear systems and their solutions, Eigen information, vector spaces, inner product spaces, and linear transformations.For individuals who want to learn abstract concepts and deal with a wide variety of applications that can be drawn from fields such as physics, chemistry, biology, geology, economics, engineering, computer science, psychology, and sociology. ... Read more

Book Description

A mathematically sophisticated treatment of matrix algebra that covers both theoretical foundations and significant applications. Ideal for students with some familiarity with vectors and matrices. The text provides several optional sections devoted to MATLAB as a computational tool, with exercises for computer solution. ... Read more

Reviews (1)

It fits for students of engineering, but...
I'm a student in EE dep of Seoul natl univ in korea, and have taken a class using this book, named 'Engineering mathematics 3' which deals linear algebra. The author introduced a little bit practical topics, rather than purely mathematical concepts. Thus one can easily trace what he wanna tell just by solving the solved-problems in text even without reading the contents. (And also in my oppinion, this book really contains lots of errors, so you must be careful of that.) When you've not got enough time to familiar with this subject, then I recommand this. If not, walk around and pick another text. I hope this review might be helpful. Thanx for reading. ... Read more

Book Description

A matroid is an abstract mathematical structure that captures combinatorial properties of matrices. This book offers a unique introduction to matroid theory, emphasizing motivations from matrix theory and applications to systems analysis. This book serves also as a comprehensive presentation of the theory and application of mixed matrices, developed primarily by the present author in the last decade. A mixed matrix is a convenient mathematical tool for systems analysis, compatible with the physical observation that "fixed constants" and "system parameters" are to be distinguished in the description of engineering systems. This book will be extremely useful to graduate students and researchers in engineering, mathematics and computer science. ... Read more

Book Description

The aim of this book is to concisely present fundamental ideas, results, and techniques in linear algebra and mainly matrix theory. The book contains eight chapters covering various topics ranging from similarity and special types of matrices to Schur complements and matrix normality. Each chapter focuses on the results, techniques, and methods that are beautiful, interesting, and representative, followed by carefully selected problems. For many theorems several different proofs are given. The book can be used as a text or a supplement for a linear algebra and matrix theory class or seminar for senior or graduate students. The only prerequisites are a decent background in elementary linear algebra and calculus. The book can also serve as a reference for instructors and researchers in the fields of algebra, matrix analysis, operator theory, statistics, computer science, engineering, operations research, economics, and other fields. (Solutions manual available for qualified instructors upon request.) ... Read more

Reviews (1)

Have you seen a matrix inequality lately?
Tons of matrix inequalities; lots of tricks. A very interesting book, indeed...... ... Read more

Book Description

This introduction to recent developments in algebraic combinatorics illustrates how research in mathematics actually progresses.The author recounts the dramatic search for and discovery of a proof of a counting formula conjectured in the late 1970s: the number of n x n alternating sign matrices, objects that generalize permutation matrices.While it was apparent that the conjecture must be true, the proof was elusive. As a result, researchers became drawn to this problem and made connections to aspects of the invariant theory of Jacobi, Sylvester, Cayley, MacMahon, Schur, and Young; to partitions and plane partitions; to symmetric functions; to hypergeometric and basic hypergeometric series; and, finally, to the six-vertex model of statistical mechanics. This volume is accessible to anyone with a knowledge of linear algebra, and it includes extensive exercises and Mathematica programs to help facilitate personal exploration. Students will learn what mathematicians actually do in an interesting and new area of mathematics, and even researchers in combinatorics will find something unique within Proofs and Confirmations. ... Read more

Reviews (3)

Fantastic!
Everyone wants to know what mathematicians do should read this book. Yes, only very few among people can get familiar or endure gorgeous complicated formulas here, But even scan or skim this book one will learn much more than he predicted. The goal of this book is simple enough: To prove a conjecture (surely when proved, it becomes a theorem) But the structure and details are well selected and carefully designed. First it introduces and explains the original problem, than talk about it's history, how it does connect with other branches of math, what these branches is about, and what the other mathematicians do on this conjecture, and at last the most exciting: how these results all bringed together and then solve this 20-year-long famous conjecture in combinatorics. There are exercises after every chapter. so this is NOT merely a history survey (as most books do). Indeed it is a textbook and contains some very excellent introductions to some branches(for example, plane partition). And I think this is THE correct style to populating mathematics--- Do not afraid of formulas, just show readers the signs, the terrible formulas, show them how and what mathematicians study and think! Every student major in mathematics should read it, and it surely is a must-have for reseachers in combinatorics.

Amazing book on how mathematical research is done
This book tells the story of one of the more important areas of research in algebraic combinatorics in the last few years. Not only does it tell the story of how the relevant conjectures were proposed, and how they were eventually proved (including some of the blind alleys and several cases where surprising connections were discovered), but it also contains a beautiful exposition of the relevant mathematics, at a level which a reader with no more than a reasonable background in linear algebra can follow. The history is fascinating and the mathematics is beautiful. If you want to know what research in mathematics is really like, this is the book to read.

Massterpiece of Mathematical Exposition
Dave Bressoud tells, in his wonderful gripping style, the fascinating recent history of the proof of the alternating sign matrix conjecture, and the not-so-recent background that lead to it. The book can be read on many levels, and is full of fascinating historical tidbits. This book is a must to anyone who wants to know how math is actually done, and who wants to share in the excitement of discovery. ... Read more

Book Description

This book provides an incisive, basic introduction to many-valued logics and to the constructions that are "many-valued" at their origin. Using the matrix method, the author sheds light on the profound problems of many-valuedness criteria and its classical characterizations.The book also includes information concerning the main systems of many-valued logic, related axiomatic constructions, and conceptions inspired by many-valuedness. With its selective bibliography and many useful historical references, this book provides logicians, computer scientists, philosophers, and mathematicians with a valuable survey of the subject. ... Read more

Reviews (7)

Unique approach of many topics, but not much 3D
For many topics, this book provides more thorough coverage for beginners than other books, and is a good resource for building up your intuition about vectors and matrices. Especially good is his discussion of matrices and operations on matrices such as gaussian elimination.

A few minor things I didn't like:

1. The whole book has a slightly "mathematical" slant, as opposed to a "geometric" slant. In other words, contrary to the title, this book is actually more about linear algebra (pure mathematics) than about geometry. For example, solving systems of equations, gaussian elimination, and the like, really don't have anything to do with geometry. Likewise, the notation is more "mathematical" than "geometric" - using e1, e2, and e3 for the basis vectors rather than x, y, and z like everybody else.

2. The book covers many topics very well in 2D - the problem is that it doesn't cover much in 3D. Some topics, of course, extend naturally from 2D into 3D and so detailed discussion isn't necessary. Other's topics dont. For example, orientation in 3D, left-handed vs. right-handed coordinate spaces, perspective projection and homegenous coordinates, quaternions. Coverage of these topics would have added a lot.

3. Other people seem to like the diagrams, but I didn't think they were that good. I think a better way to describe the diagrams is that the book has *more* diagrams than most other books, but not necessarily better ones. I personally don't like hand-drawn illustrations. And 3D diagrams needs to be rendering using shading and perspective foreshortening - schemtic-style isometric diagrams are difficult to interpret. Another example, all of the elementary geometric transformations were discussed by showing the effect of the transformation on an object. This is wonderful - most books don't do this! The only problem is that the object he choses to use is a confusing-looking circle thingy. Using a very simple object, such as a teapot would have been much better.

All-in-all, this book has some unique coverage and I would recommend it, especially for the discussion of matrices and transformations, and nested coordiante spaces. The books tends to spend time on more "purely mathematical" subject matter, which is not a bad thing, just a warning. The information on 3D topics is conspicuously lean, which is somewhat of a negative. However, I was pleased with my purchase and was able to look at several things from a different perspective.

been looking for this
good focused coverage

Best math book for graphics programmers I've found
I've read quite a few math texts looking for an understanding of the math necessary for 3D graphics software. This was the first one that I was able to read straight through, like a novel. I highly recommend it for an understanding of 3D math. It has clear descriptions that build gradually.

Good introduction to matrices for graphics
I teach computer graphics and this is the best treatment of transformation matrices for geometry I have seen. It is a little jewel. It also has a nice treatment of basic linear algebra in terms of geometric operations. My favorite part was the discussion of Cramer's rule and Gaussian Elimination in terms of geometry. A wonderful read, and I found the figures to be very helpful.

this book is terrible
We used this book for the freshmen Linear Algebra and Geometry course at my school. If it wasn't for the teacher's lectures I would not have understood any of the content in this book at all. Most of the figures are thrown in to waste ink, confuse the reader, or (in very rare occasions) represent an equation graphically. This should not under any circumstances be used as a classroom textbook (or as a reference book, etc.). ... Read more