# Matrix Algebra From a Statistician's Perspective

Springer Science & Business Media, 27.06.2008 - 634 Seiten
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 Urheberrecht

### 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.