Linear Algebra and Linear Models

A rigorous and concise introduction to linear algebra, this fully revised and compact third edition adopts the matrix theory perspective so well suited to statistical applications. It covers a host of key topics including rank additivity and singular values.

Linear Algebra and Linear Models comprises a concise and rigorous introduction to linear algebra required for statistics followed by the basic aspects of the theory of linear estimation and hypothesis testing. The emphasis is on the approach using generalized inverses. Topics such as the multivariate normal distribution and distribution of quadratic forms are included.

For this third edition, the material has been reorganised to develop the linear algebra in the first six chapters, to serve as a first course on linear algebra that is especially suitable for students of statistics or for those looking for a matrix theoretic approach to the subject. Other key features include:

coverage of topics such as rank additivity, inequalities for eigenvalues and singular values;

a new chapter on linear mixed models;

over seventy additional problems on rank: the matrix rank is an important and rich topic with connections to many aspects of linear algebra such as generalized inverses, idempotent matrices and partitioned matrices.

This text is aimed primarily at advanced undergraduate and first-year graduate students taking courses in linear algebra, linear models, multivariate analysis and design of experiments. A wealth of exercises, complete with hints and solutions, help to consolidate understanding. Researchers in mathematics and statistics will also find the book a useful source of results and problems.

Provides a concise and rigorous introduction to linear algebra from the matrix theory viewpoint which is well-suited for statistical applications Offers a compact introduction to estimation and testing in linear models covering the basic results required for further studies in linear models, multivariate analysis and design of experiments Contains a large number of exercises, including over seventy five problems on rank, with hints and solutions Includes supplementary material: sn.pub/extras

Contenu

Vector Spaces and Subspaces.- Rank, Inner Product and Nonsingularity.- Eigenvalues and Positive Definite Matrices.- Generalized Inverses.- Inequalities for Eigenvalues and Singular Values.- Rank Additivity and Matrix Partial Orders.- Linear Estimation.- Tests of Linear Hypotheses.- Linear Mixed Models.- Miscellaneous Topics.- Additional Exercises on Rank.