Matrix Algebra

1. Front Matter

   Pages i-xxix

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2. Linear Algebra

   1. Front Matter

      Pages 1-1

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   2. Basic Vector/Matrix Structure and Notation

      • James E. Gentle

      Pages 3-10

   3. Vectors and Vector Spaces

      • James E. Gentle

      Pages 11-54

   4. Basic Properties of Matrices

      • James E. Gentle

      Pages 55-183

   5. Vector/Matrix Derivatives and Integrals

      • James E. Gentle

      Pages 185-225

   6. Matrix Transformations and Factorizations

      • James E. Gentle

      Pages 227-263

   7. Solution of Linear Systems

      • James E. Gentle

      Pages 265-306

   8. Evaluation of Eigenvalues and Eigenvectors

      • James E. Gentle

      Pages 307-325

3. Applications in Data Analysis

   1. Front Matter

      Pages 327-327

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   2. Special Matrices and Operations Useful in Modeling and Data Analysis

      • James E. Gentle

      Pages 329-398

   3. Selected Applications in Statistics

      • James E. Gentle

      Pages 399-458

4. Numerical Methods and Software

   1. Front Matter

      Pages 459-459

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   2. Numerical Methods

      • James E. Gentle

      Pages 461-521

   3. Numerical Linear Algebra

      • James E. Gentle

      Pages 523-538

   4. Software for Numerical Linear Algebra

      • James E. Gentle

      Pages 539-585

5. Back Matter

   Pages 587-648

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This textbook for graduate and advanced undergraduate students presents the theory of matrix algebra for statistical applications, explores various types of matrices encountered in statistics, and covers numerical linear algebra. Matrix algebra is one of the most important areas of mathematics in data science and in statistical theory, and the second edition of this very popular textbook provides essential updates and comprehensive coverage on critical topics in mathematics in data science and in statistical theory.

Part I offers a self-contained description of relevant aspects of the theory of matrix algebra for applications in statistics. It begins with fundamental concepts of vectors and vector spaces; covers basic algebraic properties of matrices and analytic properties of vectors and matrices in multivariate calculus; and concludes with a discussion on operations on matrices in solutions of linear systems and in eigenanalysis. Part II considers various types of matricesencountered in statistics, such as projection matrices and positive definite matrices, and describes special properties of those matrices; and describes various applications of matrix theory in statistics, including linear models, multivariate analysis, and stochastic processes. Part III covers numerical linear algebra—one of the most important subjects in the field of statistical computing. It begins with a discussion of the basics of numerical computations and goes on to describe accurate and efficient algorithms for factoring matrices, how to solve linear systems of equations, and the extraction of eigenvalues and eigenvectors.

Although the book is not tied to any particular software system, it describes and gives examples of the use of modern computer software for numerical linear algebra. This part is essentially self-contained, although it assumes some ability to program in Fortran or C and/or the ability to use R or Matlab.

The first two parts of the text are ideal for a course in matrix algebra for statistics students or as a supplementary text for various courses in linear models or multivariate statistics. The third part is ideal for use as a text for a course in statistical computing or as a supplementary text for various courses that emphasize computations.

New to this edition

• 100 pages of additional material

• 30 more exercises—186 exercises overall
• Added discussion of vectors and matrices with complex elements
• Additional material on statistical applications
• Extensive and reader-friendly cross references and index

• matrix
• linear algebra
• numerical analysis
• optimization
• linear model
• vector
• linear transformation
• singular value decomposition
• generalized inverse
• determinant
• eigenvalue
• eigenvector
• graph theory
• linear system
• positive definite
• R software
• vector space
• inner product
• geometry

“Gentle has put in a lot of time and effort to writing this book with careful attention to details. … it is all needed to make sure the student has a firm and solid understanding of matrix algebra on the graduate level. I would recommend this book for all those who teach graduate level matrix algebra or … to those undergraduate students who wish to have an independent study.” (Peter Olszewski, MAA Reviews, January, 2018)

“Beautifully written, easy to read, with a well subindexed index of 16 pages and a bibliography of 13 that includes most modern and relevant textbooks and articles in the area of matrix theory and computations, as well as for statistics and big data computations.” (Frank Uhlig, zbMATH 1386.15002, 2018)

“This very reader-friendly written volume presents an opportunity to graduate students and researchers to enjoy reading on the classical matrix analysis in its modern applications to statistics and to implement thesemethods in practical problem solving.” (Stan Lipovetsky, Technometrics, Vol. 60 (2), 2018)

• Fairfax, USA

  James E. Gentle

​James E. Gentle, PhD, is University Professor of Computational Statistics at George Mason University. He is a Fellow of the American Statistical Association (ASA) and of the American Association for the Advancement of  Science. Professor Gentle has held several national offices in the ASA and has served as editor and associate editor of journals of the ASA as well as for other journals in statistics and computing. He is author of Random Number Generation and Monte Carlo Methods (Springer, 2003) and Computational Statistics (Springer, 2009).

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