Umfang:
Online-Ressource (VIII, 167p. 1 illus, digital)
Ausgabe:
3rd ed. 2012
ISBN:
9781447127390
Serie:
Universitext
Inhalt:
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.
Inhalt:
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.
Anmerkung:
Description based upon print version of record
,
""Linear Algebra and Linear Models""; ""Preface""; ""About the Second Edition""; ""About the Third Edition""; ""Contents""; ""Chapter 1: Vector Spaces and Subspaces""; ""1.1 Preliminaries""; ""1.2 Vector Spaces""; ""1.3 Basis and Dimension""; ""1.4 Exercises""; ""Chapter 2: Rank, Inner Product and Nonsingularity""; ""2.1 Rank""; ""2.2 Inner Product""; ""2.3 Nonsingularity""; ""2.4 Frobenius Inequality""; ""2.5 Exercises""; ""Chapter 3: Eigenvalues and Positive De nite Matrices""; ""3.1 Preliminaries""; ""3.2 The Spectral Theorem""; ""3.3 Schur Complement""; ""3.4 Exercises""
,
""Chapter 4: Generalized Inverses""""4.1 Preliminaries""; ""4.2 Minimum Norm and Least Squares g-Inverse""; ""4.3 Moore-Penrose Inverse""; ""4.4 Exercises""; ""Chapter 5: Inequalities for Eigenvalues and Singular Values""; ""5.1 Eigenvalues of a Symmetric Matrix""; ""5.2 Singular Values""; ""5.3 Minimax Principle and Interlacing""; ""5.4 Majorization""; ""5.5 Volume of a Matrix""; ""5.6 Exercises""; ""Chapter 6: Rank Additivity and Matrix Partial Orders""; ""6.1 Characterizations of Rank Additivity""; ""6.2 The Star Order""; ""6.3 Exercises""; ""Chapter 7: Linear Estimation""
,
""7.1 Linear Model""""7.2 Estimability""; ""7.3 Residual Sum of Squares""; ""One-Way Classi cation""; ""7.4 General Linear Model""; ""7.5 Exercises""; ""Chapter 8: Tests of Linear Hypotheses""; ""8.1 Multivariate Normal Distribution""; ""8.2 Quadratic Forms and Cochran's Theorem""; ""8.3 One-Way and Two-Way Classi cations""; ""Two-Way Classi cation with Interaction""; ""8.4 Linear Hypotheses""; ""Maximum Likelihood Estimates of beta and sigma2""; ""8.5 Multiple Correlation""; ""8.6 Exercises""; ""Chapter 9: Linear Mixed Models""; ""9.1 Fixed Effects and Random Effects""
,
""9.2 ML and REML Estimators""""9.3 ANOVA Estimators""; ""9.4 Prediction of Random Effects""; ""9.5 Exercises""; ""Chapter 10: Miscellaneous Topics""; ""10.1 Principal Components""; ""10.2 Canonical Correlations""; ""10.3 Reduced Normal Equations""; ""10.4 The C-Matrix""; ""10.5 E-, A- and D-Optimality""; ""10.6 Exercises""; ""Chapter 11: Additional Exercises on Rank""; ""Chapter 12: Hints and Solutions to Selected Exercises""; ""Chapter 1""; ""Chapter 2""; ""Chapter 3""; ""Chapter 4""; ""Chapter 5""; ""Chapter 6""; ""Chapter 7""; ""Chapter 8""; ""Chapter 9""; ""Chapter 10""; ""Chapter 11""
,
""Chapter 13: Notes""""Chapter 1""; ""Chapter 2""; ""Chapter 3""; ""Chapter 4""; ""Chapter 5""; ""Chapter 6""; ""Chapter 7""; ""Chapter 8""; ""Chapter 9""; ""Chapter 10""; ""Chapter 11""; ""References""; ""Index""
Weitere Ausg.:
ISBN 9781447127383
Weitere Ausg.:
Buchausg. u.d.T. Bapat, Ravi B. Linear algebra and linear models New Delhi : Hindustan Book Agency, 2012 ISBN 9789380250281
Weitere Ausg.:
ISBN 9781447127383
Weitere Ausg.:
ISBN 9781447127390
Sprache:
Englisch
Fachgebiete:
Mathematik
Schlagwort(e):
Lineare Algebra
;
Lineares Modell
;
Lineare Algebra
;
Lineares Modell
DOI:
10.1007/978-1-4471-2739-0
URL:
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