1 comments about Elementary Matrix Algebra (Dover Books on Mathematics).

1. The definitions and derivations are clear and carefully explained. The text starts with the basics, and moves at a brisk and comfortable pace to quite advanced matrix operations and properties. Quite a lot of numerical examples are given throughout the text as examples/counter-examples to clarify misconceptions and surprising properties of a matrix (such as the presence of divisors of zero). There are a large number of exercises included, and the later ones in a section are usually quite challenging and enlightening; many of them extend the main text substantially without increasing the length of the exposition. Some of the matrix and vector space notations used are dated (circa 1973). But otherwise this is a great text to get up to speed with and to lay a solid foundation for more advanced matrix texts like Gantmacher and Horn/Johnson.

1 comments about The Theory of Matrices in Numerical Analysis (Dover Books on Mathematics).

1. Table of contents
   
   1. SOME BASIC IDENTITIES AND INEQUALITIES
   1.0 Objectives; Notation
   1.1 Elementary Matrices
   1.2 Some Factorizations
   1.3 Projections, and the General Reciprocal
   1.4 Some Determinantal Identities
   1.5 Lanczos Algorithm for Tridiagonalization
   1.6 Orthogonal Polynomials
   References
   Problems and Exercises
   
   2. NORMS, BOUNDS, AND CONVERGENCE
   2.0 The Notion of a Norm
   2.1 Convex Sets and Convex Bodies
   2.2 Norms and Bounds
   2.3 Norms, Bounds, and Spectral Radii
   2.4 Nonnegative Matrices
   2.5 Convergence; Functions of Matrices
   References
   Problems and Exercises
   
   3. LOCALIZATION THEOREMS AND OTHER INEQUALITIES
   3.0 Basic Definitions
   3.1 Exclusion Theorems
   3.2 Inclusion and Separation Theorems
   3.3 Minimax Theorems and the Field of Values
   3.4 Inequalities of Wielandt Kantorovich
   References
   Problems and Exercises
   
   4. THE SOLUTION OF LINEAR SYSTEMS: METHODS OF SUCCESSIVE APPROXIMATION
   4.0 Direct Methods and Others
   4.1 The Inversion of Matrices
   4.2 Methods of Projection
   4.3 Norm-Reducing Methods
   References
   Problems and Exercises
   
   5. DIRECT METHODS OF INVERSION
   5.0 Uses of the Inverse
   5.1 The Method of Modification
   5.2 Triangularization
   5.3 A More General Formulation
   5.4 Orthogonal Triangularization
   5.5 Orthogonalization
   5.6 Orthogonalization and Projection
   5.7 The Method of Conjugate Gradients
   References
   Problems and Exercises
   
   6. PROPER VALUES AND VECTORS: NORMALIZATION AND REDUCTION OF THE MATRIX
   6.0 Purpose of Normalization
   6.1 The Method of Krylov
   6.2 The Weber-Voetter Method
   6.3 The Method of Danilevskii
   6.4 The Hessenberg and the Lanczos Reductions
   6.5 Proper Values and Vectors
   6.6 The Method of Samuelson and Bryan
   6.7 The Method of Leverrier
   References
   Problems and Exercises
   
   7. PROPER VALUES AND VECTORS: SUCCESSIVE APPROXIMATION
   7.0 Methods of Successive Approximation
   7.1 The Method of Jacobi
   7.2 The Method of Collar and Jahn
   7.3 Powers of a Matrix
   7.4 Simple Iteration (the Power Method)
   7.5 Multiple Roots and Principal Vectors
   7.6 Staircase Iteration (Treppeniteration)
   7.7 The LR-Transformation
   7.8 Bi-iteration
   7.9 The QR-Transformation
   References
   Problems and Exercises
   
   BIBLIOGRAPHY
   INDEX

No comments about Matrix Algebra: Theory, Computations, and Applications in Statistics (Springer Texts in Statistics).

No comments about Matrix and Tensor Calculus: With Applications to Mechanics, Elasticity and Aeronautics.

1 comments about Modern Matrix Algebra.

1. The math concepts developed in this book are used pervasively in many sciences. And indeed throughout engineering. It is impossible to envisage analysing circuits without matrices.
   
   The book carefully explains the various ideas. If you are careful, you can develop a good geometric intuition, starting with the key idea that a vector sits in Euclidean n-space. And that a matrix can be used to transform from n-space to m-space, where m and n need not be equal.
   
   Certainly, the book shows how to invert matrices; so you get to learn determinants en route. Picking up Matlab is also a useful trait. It relieves you of a lot of tedious computations. When I learnt this stuff, no symbolic algebraic program was easily available. So our problem sets were done by hand.

3 comments about Proofs and Confirmations: The Story of the Alternating-Sign Matrix Conjecture (Spectrum).

1. 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.



2. 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.



3. 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.

No comments about Nonnegative Matrices in the Mathematical Sciences (Classics in Applied Mathematics).

1 comments about Matrix Structural Analysis.

1. It is a very good, albeit succinct, reference on the direct stiffness method. Very useful for review or a quick reference. I used it as pocket TA during Prof. Filippou CE 220 course, even if the course covered a wider range of topics.

2 comments about Matrix Theory Vol. 1.

1. Anybody who wants to know the theory of matrices should keep this book on his or her desk.



2. This classical book constitutes, even 60 years after been written, one of the most valuable book on matrix theory, written by eminent soviet mathematician Felix R. Gantmacher. In this book many aspects are covered, from the general theory of linear spaces (chapters 1-10), numerical methods for calculation of characteristic polynomials), applications to study of differential equations, Markov processes or applications to elastic systems. Many proof are given in alternative way to usual books, and are interesting variations. Anyone interested in matrices and applications will find here some subject of interest. It is really one of the best books on matrices which has been written.
   {the comments refer to the original Nauka edition of 1966, but also apply to the translations and reprints made of it}

2 comments about Matrix Analysis for Statistics (Wiley Series in Probability and Statistics).

1. 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?




2. If queried, the mathematician that does not work in statistics will most likely not feel that there is a need for a separate course in matrices applied to statistics. There is no doubt that statisticians need to know a great deal about matrices, so the question comes down to whether the traditional math courses in linear algebra are sufficient. In either case, it is still of benefit to have one source that statisticians can consult as a reference for problems with matrices and this book can serve as that source.
   Mathematicians with little experience in statistics will have no difficulty in understanding the contents of the book, as nearly all of it is mathematical rather than statistical in nature. The first three chapters are:
   
   *) A review of elementary matrix algebra.
   *) Vector spaces.
   *) Eigenvalues and eigenvectors.
   
   With the exception of ten pages devoted to random vectors in chapter 1, there were few major items gleaned from statistics. They could serve as the first three chapters of any book on matrix operations. While the remaining chapters do contain more statistical concepts, the overwhelming majority of the material does not involve problems in statistics. There are a large number of problems at the end of the chapters and solutions are not provided.
   The remaining chapters are:
   
   *) Matrix factorizations and matrix norms.
   *) Generalized inverses.
   *) Systems of linear equations.
   *) Partitioned matrices.
   *) Special matrices and matrix products.
   *) Matrix derivatives and related topics.
   *) Some special topics related to quadratic forms.
   
   The level of difficulty is within the reach of an advanced undergraduate, although it was written for graduate students in statistics. In my opinion, previous experience in statistics would be helpful, but not required for a reader to understand the material. If you are looking for a book on matrix operations, this one will serve your purpose, independent of whether your focus is on statistics.
   
   Published in Journal of Recreational Mathematics, reprinted with permission.