Matrix Algebra From a Statistician's Perspective

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Springer Science & Business Media, 27.06.2008 - 634 Seiten
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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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Inhalt

Matrices
1
12 Basic Operations
2
13 Some Basic Types of Matrices
5
Exercises
10
Submatrices and Partitioned Matrices
13
23 Some Results on the Product of a Matrix and a Column Vector
19
24 Expansion of a Matrix in Terms of Its Rows Columns or Elements
20
Linear Dependence and Independence
23
154 Differentiation of Matrix Sums Products and Transposes and of Matrices of Constants
300
155 Differentiation of a Vector or Unrestricted or Symmetric Matrix With Respect to Its Elements
303
156 Differentiation of a Trace of a Matrix
304
157 The Chain Rule
306
158 FirstOrder Partial Derivatives of Determinants and Inverse and Adjoint Matrices
308
159 SecondOrder Partial Derivatives of Determinants and Inverse Matrices
312
1510 Differentiation of Generalized Inverses
314
1511 Differentiation of Projection Matrices
319

Linear Spaces Row and Column Spaces
27
42 Subspaces
29
43 Bases
31
44 Rank of a Matrix
36
45 Some Basic Results on Partitioned Matrices and on Sums of Matrices
41
Exercises Section 41
46
Trace of a Square Matrix
49
52 Trace of a Product
50
53 Some Equivalent Conditions
52
Geometrical Considerations
54
62 Orthogonal and Orthonormal Sets
61
63 Schwarz Inequality
62
64 Orthonormal Bases
63
Exercises
68
Linear Systems Consistency and Compatibility
71
72 Consistency
72
73 Compatibility
73
Exercise
77
Inverse Matrices
79
82 Properties of Inverse Matrices
81
83 Premultiplication or Postmultiplication by a Matrix of Full Column or Row Rank
82
84 Orthogonal Matrices
84
85 Some Basic Results on the Ranks and Inverses of Partitioned Matrices
88
Exercises
103
Generalized Inverses
106
92 Some Alternative Characterizations
109
93 Some Elementary Properties
117
94 Invariance to the Choice of a Generalized Inverse
119
95 A Necessary and Sufficient Condition for the Consistency of aLinearSystem
120
96 Some Results on the Ranks and Generalized Inverses of Partitioned Matrices
121
97 Extension of Some Results on Systems of the Form AX D B to Systems of the Form AXC D B
126
Idempotent Matrices
133
102 Some Basic Results
134
Exercises
136
Linear Systems Solutions
139
112 General Form of a Solution a Homogeneous linear systems
140
113 Number of Solutions
142
115 An Alternative Expression for the General Form of a Solution
144
116 Equivalent Linear Systems
145
117 Null and Column Spaces of Idempotent Matrices
146
119 A Computational Approach
149
1110 Linear Combinations of the Unknowns
150
1111 Absorption
152
1112 Extensions to Systems of the Form AXC D B
157
Projections and Projection Matrices
161
122 Projection of a Column Vector
163
123 Projection Matrices
166
125 Orthogonal Complements
172
Determinants
179
132 Some Basic Properties of Determinants
183
133 Partitioned Matrices Products of Matrices and Inverse Matrices
187
134 A Computational Approach
191
136 Vandermonde Matrices
195
137 Some Results on the Determinant of the Sum of Two Matrices
197
138 Laplaces Theorem and the BinetCauchy Formula
200
Exercises Section 131
205
Linear Bilinear and Quadratic Forms
209
142 Nonnegative Definite Quadratic Forms and Matrices a Definitions
212
143 Decomposition of Symmetric and Symmetric Nonnegative Definite Matrices
218
144 Generalized Inverses of Symmetric Nonnegative Definite Matrices
222
145 LDU U0DU and Cholesky Decompositions
223
146 SkewSymmetric Matrices
239
147 Trace of a Nonnegative Definite Matrix a Basic results
240
148 Partitioned Nonnegative Definite Matrices
243
149 Some Results on Determinants
247
1410 Geometrical Considerations
255
1411 Some Results on Ranks and Row and Column Spaces and on Linear Systems
259
1412 Projections Projection Matrices and Orthogonal Complements
260
Exercises Section 141
277
Matrix Differentiation
289
151 Definitions Notation and Other Preliminaries
290
Some Elementary Results
296
153 Differentiation of Linear and Quadratic Forms
298
1512 Evaluation of Some Multiple Integrals
324
Exercises Section 151
327
Bibliographic and Supplementary Notes
335
Kronecker Products and the Vec and Vech Operators
336
Definition and Some Basic Properties
343
163 VecPermutation Matrix
347
164 The Vech Operator
354
165 Reformulation of a Linear System
367
166 Some Results on Jacobian Matrices
368
Exercises
371
Bibliographic and Supplementary Notes
377
Intersections and Sums of Subspaces
379
172 Some Results on Row and Column Spaces and on the Ranks of Partitioned Matrices
385
173 Some Results on Linear Systems and on Generalized Inverses of Partitioned Matrices
392
Sum of Their Dimensions Versus Dimension of Their Sum
396
175 Some Results on the Rank of a Product of Matrices a Initial presentation of results
398
176 Projections Along a Subspace a Some general results and terminology
402
177 Some Further Results on the Essential Disjointness and Orthogonality of Subspaces and on Projections and Projection Matrices
409
Exercises
411
Bibliographic and Supplementary Notes
417
Sums and Differences of Matrices
419
182 Some Results on Inverses and Generalized Inverses and on Linear Systems
423
183 Some Results on Positive and Nonnegative Definiteness
437
184 Some Results on Idempotency a Basic results
439
185 Some Results on Ranks
444
Exercises
450
Bibliographic and Supplementary Notes
458
Minimization of a SecondDegree Polynomial in n Variables Subject to Linear Constraints
459
191 Unconstrained Minimization
460
192 Constrained Minimization a Basic results
463
193 Explicit Expressions for Solutions to the Constrained Minimization Problem
468
194 Some Results on Generalized Inverses of Partitioned Matrices
476
195 Some Additional Results on the Form of Solutions to the Constrained Minimization Problem
483
196 Transformation of the Constrained Minimization Problem to an Unconstrained Minimization Problem
489
197 The Effect of Constraints on the Generalized Least Squares Problem
491
Bibliographic and Supplementary Notes
495
The MoorePenrose Inverse
496
202 Some Special Cases
499
203 Special Types of Generalized Inverses
500
204 Some Alternative Representations and Characterizations
507
205 Some Basic Properties and Relationships
508
206 Minimum Norm Solution to the Least Squares Problem
512
208 Differentiation of the MoorePenrose Inverse
514
Exercises
517
Bibliographic and Supplementary Notes
519
Eigenvalues and Eigenvectors
521
211 Definitions Terminology and Some Basic Results
522
212 Eigenvalues of Triangular or BlockTriangular Matrices and of Diagonal or BlockDiagonal Matrices
528
213 Similar Matrices
530
214 Linear Independence of Eigenvectors
534
215 Diagonalization
537
216 Expressions for the Trace and Determinant of a Matrix
545
217 Some Results on the MoorePenrose Inverse of a Symmetric Matrix
546
218 Eigenvalues of Orthogonal Idempotent and Nonnegative Definite Matrices
548
219 Square Root of a Symmetric Nonnegative Definite Matrix
550
2110 Some Relationships
551
2111 Eigenvalues and Eigenvectors of Kronecker Products of Square Matrices
554
2112 Singular Value Decomposition
556
2113 Simultaneous Diagonalization
566
2114 Generalized Eigenvalue Problem
569
2115 Differentiation of Eigenvalues and Eigenvectors
571
2116 An Equivalence Involving Determinants and Polynomials
574
Some Properties of Polynomials in a Single Variable
580
Bibliographic and Supplementary Notes
588
Linear Transformations
589
224 Matrix Representation of a Linear Transformation a Background definition and some basic properties
601
225 Terminology and Properties Shared by a Linear Transformation and Its Matrix Representation
609
226 Linear Functionals and Dual Transformations
612
Exercises Section 221
616
References
621
Index
624
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Über den Autor (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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