Matrix Algebra From a Statistician's Perspective

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Springer Science & Business Media, Jun 27, 2008 - Mathematics - 634 pages
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Matrix algebra plays a very important role in statistics and in many other dis- plines. In many areas of statistics, it has become routine to use matrix algebra in thepresentationandthederivationorveri?cationofresults. Onesuchareaislinear statistical models; another is multivariate analysis. In these areas, a knowledge of matrix algebra isneeded in applying important concepts, as well as instudying the underlying theory, and is even needed to use various software packages (if they are to be used with con?dence and competence). On many occasions, I have taught graduate-level courses in linear statistical models. Typically, the prerequisites for such courses include an introductory (- dergraduate) course in matrix (or linear) algebra. Also typically, the preparation provided by this prerequisite course is not fully adequate. There are several r- sons for this. The level of abstraction or generality in the matrix (or linear) algebra course may have been so high that it did not lead to a “working knowledge” of the subject, or, at the other extreme, the course may have emphasized computations at the expense of fundamental concepts. Further, the content of introductory courses on matrix (or linear) algebra varies widely from institution to institution and from instructor to instructor. Topics such as quadratic forms, partitioned matrices, and generalized inverses that play an important role in the study of linear statistical models may be covered inadequately if at all.
 

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Contents

Matrices
1
Submatrices and Partitioned Matrices
13
Linear Dependence and Independence
23
Trace of a Square Matrix
49
Consistency and Compatibility
71
Inverse Matrices
79
Generalized Inverses 107
106
Idempotent Matrices
133
Linear Bilinear and Quadratic Forms
209
Matrix Differentiation
289
Kronecker Products and the Vec and Vech Operators 337
336
Intersections and Sums of Subspaces
379
Sums and Differences of Matrices
419
Minimization of a SecondDegree Polynomial in n Variables
459
The MoorePenrose Inverse 497
496
Eigenvalues and Eigenvectors
521

Solutions
139
Projections and Projection Matrices
161
Determinants
179
Linear Transformations
589
References
621
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About the author (2008)

David A. Harville is a research staff member in the Mathematical Sciences Department of the IBM T.J.Watson Research Center. Prior to joining the Research Center he spent ten years as a mathematical statistician in the Applied Mathematics Research Laboratory of the Aerospace Research Laboratories (at Wright-Patterson, FB, Ohio, followed by twenty years as a full professor in the Department of Statistics at Iowa State University. He has extensive experience in the area of linear statistical models, having taught (on numberous occasions) M.S.and Ph.D.level courses on that topic,having been the thesis adviser of 10 Ph.D. students,and having authored over 60 research articles. His work has been recognized by his election as a Fellow of the American Statistical Association and the Institute of Mathematical Statistics and as a member of the International Statistical Institute and by his having served as an associate editor of Biometrics and of the Journal of the American Statistical Association.

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