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Matrix Algebra Useful for Statistics von Searle, Shayle R. (eBook)

  • Erscheinungsdatum: 10.04.2017
  • Verlag: Wiley
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Matrix Algebra Useful for Statistics

A thoroughly updated guide to matrix algebra and it uses in statistical analysis and features SAS®, MATLAB®, and R throughout This Second Edition addresses matrix algebra that is useful in the statistical analysis of data as well as within statistics as a whole. The material is presented in an explanatory style rather than a formal theorem-proof format and is self-contained. Featuring numerous applied illustrations, numerical examples, and exercises, the book has been updated to include the use of SAS, MATLAB, and R for the execution of matrix computations. In addition, André I. Khuri, who has extensive research and teaching experience in the field, joins this new edition as co-author. The Second Edition also: Contains new coverage on vector spaces and linear transformations and discusses computational aspects of matrices Covers the analysis of balanced linear models using direct products of matrices Analyzes multiresponse linear models where several responses can be of interest Includes extensive use of SAS, MATLAB, and R throughout Contains over 400 examples and exercises to reinforce understanding along with select solutions Includes plentiful new illustrations depicting the importance of geometry as well as historical interludes
Matrix Algebra Useful for Statistics, Second Edition is an ideal textbook for advanced undergraduate and first-year graduate level courses in statistics and other related disciplines. The book is also appropriate as a reference for independent readers who use statistics and wish to improve their knowledge of matrix algebra. THE LATE SHAYLE R. SEARLE, PHD, was professor emeritus of biometry at Cornell University. He was the author of Linear Models for Unbalanced Data and Linear Models and co-author of Generalized, Linear, and Mixed Models, Second Edition, Matrix Algebra for Applied Economics, and Variance Components, all published by Wiley. Dr. Searle received the Alexander von Humboldt Senior Scientist Award, and he was an honorary fellow of the Royal Society of New Zealand. ANDRÉ I. KHURI, PHD, is Professor Emeritus of Statistics at the University of Florida. He is the author of Advanced Calculus with Applications in Statistics, Second Edition and co-author of Statistical Tests for Mixed Linear Models, all published by Wiley. Dr. Khuri is a member of numerous academic associations, among them the American Statistical Association and the Institute of Mathematical Statistics. The late Shayle R. Searle, PhD, was professor emeritus of biometry at Cornell University. He was the author of Linear Models for Unbalanced Data and Linear Models and co-author of Generalized, Linear, and Mixed Models, Second Edition, Matrix Algebra for Applied Economics, and Variance Components, all published by Wiley. Dr. Searle received the Alexander von Humboldt Senior Scientist Award, and he was an honorary fellow of the Royal Society of New Zealand. André I. Khuri, PhD, is Professor Emeritus of Statistics at the University of Florida. He is the author of Advanced Calculus with Applications in Statistics, Second Edition and co-author of Statistical Tests for Mixed Linear Models, all published by Wiley. Dr. Khuri is a member of numerous academic associations, among them the American Statistical Association and the Institute of Mathematical Statistics.

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Matrix Algebra Useful for Statistics

Introduction

Historical Perspectives on Matrix Algebra

It is difficult to determine the origin of matrices from the historical point of view. Given the association between matrices and simultaneous linear equations, it can be argued that the history of matrices goes back to at least the third century BC. The Babylonians used simultaneous linear equations to study problems that pertained to agriculture in the fertile region between the Tigris and Euphrates rivers in ancient Mesopotamia (present day Iraq). They inscribed their findings, using a wedge-shaped script, on soft clay tablets which were later baked in ovens resulting in what is known as cuneiform tablets (see Figures 1 and 2).

This form of writing goes back to about 3000 BC (see Knuth, 1972). For example, a tablet dating from around 300 BC was found to contain a description of a problem that could be formulated in terms of two simultaneous linear equations in two variables. The description referred to two fields whose total area, the rate of production of grain per field, and their total yield were all given. It was required to determine the area of each field (see O'Connor and Robertson, 1996). The ancient Chinese also dealt with simultaneous linear equations between 200 BC and 100 BC in studying, for example, corn production. In fact, the text, Nine Chapters on the Mathematical Art , which was written during the Han Dynasty, played an important role in the development of mathematics in China. It was a practical handbook of mathematics consisting of 246 problems that pertained to engineering, surveying, trade, and taxation issues (see O'Connor and Robertson, 2003).

The modern development of matrices and matrix algebra did not materialize until the nineteenth century with the work of several mathematicians, including Augustin-Louis Cauchy, Ferdinand Georg Frobenius, Carl Friedrich Gauss, Arthur Cayley, and James Joseph Sylvester, among others. The use of the word "matrix" was first introduced by Sylvester in 1850. This terminology became more common after the publication of Cayley's (1858) memoir on the theory of matrices. In 1829, Cauchy gave the first valid proof that the eigenvalues of a symmetric matrix must be real. He was also instrumental in creating the theory of determinants in his 1812 memoir. Frobenius (1877) wrote an important monograph in which he provided a unifying theory of matrices that combined the work of several other mathematicians. Hawkins (1974) described Frobenius' paper as representing "an important landmark in the history of the theory of matrices." Hawkins (1975) discussed Cauchy's work and its historical significance to the consideration of algebraic eigenvalue problems during the 18th century.

Figure 0.1 A Cuneiform Tablet with 97 Linear Equations (YBC4695-1). Yale Babylonian Collection, Yale University Library, New Haven, CT.

Science historians and mathematicians have regarded Cayley as the founder of the theory of matrices. His 1858 memoir was considered "the foundation upon which other mathematicians were able to erect the edifice we now call the theory of matrices" (see Hawkins, 1974, p. 561). Cayley was interested in devising a contracted notation to represent a system of m linear equations in n variables of the form

where the aij 's are given as coefficients. Cayley and other contemporary algebraists proposed replacing the m equations with a single matrix equation such as

Figure 0.2 An Old Babylonian Mathematical Text with Linear Equations (YBC4695-2). Yale Babylonian Collection, Yale University Library, New Haven, CT.

Cayley regarded such a scheme as an operator acting upon the variables, x 1, x 2, ..., xn to produce the variables y 1, y 2, ..., ym . This is a multivariable extension of the action of the single coefficient

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