James E. Gentle

Matrix Algebra (eBook, PDF)

Theory, Computations, and Applications in Statistics

• Format: PDF

Produktbeschreibung

Matrix algebra is one of the most important areas of mathematics for data analysis and for statistical theory. The first part of this book presents the relevant aspects of the theory of matrix algebra for applications in statistics. This part begins with the fundamental concepts of vectors and vector spaces, next covers the basic algebraic properties of matrices, then describes the analytic properties of vectors and matrices in the multivariate calculus, and finally discusses operations on matrices in solutions of linear systems and in eigenanalysis. This part is essentially self-contained.

The second part of the book begins with a consideration of various types of matrices encountered in statistics, such as projection matrices and positive definite matrices, and describes the special properties of those matrices. The second part also describes some of the many applications of matrix theory in statistics, including linear models, multivariate analysis, and stochastic processes. The brief coverage in this part illustrates the matrix theory developed in the first part of the book. The first two parts of the book can be used as the text 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 of this book covers numerical linear algebra. It begins with a discussion of the basics of numerical computations, and then describes accurate and efficient algorithms for factoring matrices, solving linear systems of equations, and extracting 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/S-Plus or Matlab. This part of the book can be used as the text for a course in statistical computing, or as a supplementary text for various courses that emphasize computations.

The book includes a large number of exercises with some solutions provided in an appendix.

James E. Gentle 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. He has held several national offices in the ASA and has served as 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, Second Edition, and Elements of Computational Statistics.

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Produktdetails

• Verlag: Springer-Verlag GmbH
• Seitenzahl: 530
• Englisch
• ISBN-13: 9780387708737
• Artikelnr.: 43729679

Autorenporträt

James E. Gentle is University Professor Emeritus at George Mason University. He is a Fellow of the American Statistical Association (ASA) and of the American Association for the Advancement of Science. He has held several national offices in the ASA and has served as 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 , Computational Statistics, and Statistical Analysis of Financial Data. He is co-editor-in-chief of Wiley Interdisciplinary Reviews: Computational Statistics.

Inhaltsangabe

Part I Linear Algebra.- 1 Basic Vector/Matrix Structure and Notation.- 2 Vectors and Vector Spaces.- 3 Basic Properties of Matrices.- 4 Vector/Matrix Derivatives and Integrals.- 5 Matrix Transformations and Factorizations.- 6 Solution of Linear Systems.- 7 Evaluation of Eigenvalues and Eigenvectors.

Rezensionen

"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)