Matrix Algebra

Matrix Algebra is the first volume of the Econometric Exercises Series. It contains exercises relating to course material in matrix algebra that students are expected to know while enrolled in an (advanced) undergraduate or a postgraduate course in econometrics or statistics. The book contains a comprehensive collection of exercises, all with full answers. But the book is not just a collection of exercises; in fact, it is a textbook, though one that is organized in a completely different manner than the usual textbook. The volume can be used either as a self-contained course in matrix algebra or as a supplementary text. Cover......Page 1 Half-title......Page 3 Series-title......Page 4 Title......Page 5 Copyright......Page 6 Dedication......Page 7 Contents......Page 9 List of exercises......Page 13 Preface to the Series......Page 27 Preface......Page 31 1 Vectors......Page 33 1.1 Real vectors......Page 36 1.2 Complex vectors......Page 43 Notes......Page 45 2 Matrices......Page 47 2.1 Real matrices......Page 51 2.2 Complex matrices......Page 71 Notes......Page 74 3 Vector spaces......Page 75 3.1 Complex and real vector spaces......Page 79 3.2 Inner-product space......Page 93 Notes......Page 103 4 Rank, inverse, and determinant......Page 105 4.1 Rank......Page 107 4.2 Inverse......Page 115 4.3 Determinant......Page 119 Notes......Page 128 5 Partitioned matrices......Page 129 5.1 Basic results and multiplication relations......Page 130 5.2 Inverses......Page 135 5.3 Determinants......Page 141 5.4 Rank (in)equalities......Page 151 5.5 The sweep operator......Page 158 Notes......Page 161 6 Systems of equations......Page 163 6.1 Elementary matrices......Page 164 6.2 Echelon matrices......Page 169 6.3 Gaussian elimination......Page 175 6.4 Homogeneous equations......Page 180 6.5 Nonhomogeneous equations......Page 183 Notes......Page 186 7 Eigenvalues, eigenvectors, and factorizations......Page 187 7.1 Eigenvalues and eigenvectors......Page 190 7.2 Symmetric matrices......Page 207 7.3 Some results for triangular matrices......Page 214 7.4 Schur’s decomposition theorem and its consequences......Page 219 7.5 Jordan’s decomposition theorem......Page 224 7.6 Jordan chains and generalized eigenvectors......Page 233 Notes......Page 239 8 Positive (semi)definite and idempotent matrices......Page 241 8.1 Positive (semi)definite matrices......Page 243 8.2 Partitioning and positive (semi)definite matrices......Page 260 8.3 Idempotent matrices......Page 263 Notes......Page 274 9 Matrix functions......Page 275 9.1 Simple functions......Page 278 9.2 Jordan representation......Page 287 9.3 Matrix-polynomial representation......Page 297 Notes......Page 302 10 Kronecker product, vec-operator, and Moore-Penrose inverse......Page 305 10.1 The Kronecker product......Page 306 10.2 The vec-operator......Page 313 10.3 The Moore-Penrose inverse......Page 316 10.4 Linear vector and matrix equations......Page 324 10.5 The generalized inverse......Page 327 Notes......Page 329 11 Patterned matrices: commutation- and duplication matrix......Page 331 11.1 The commutation matrix......Page 332 11.2 The symmetrizer matrix......Page 339 11.3 The vech-operator and the duplication matrix......Page 343 11.4 Linear structures......Page 350 Notes......Page 352 12 Matrix inequalities......Page 353 12.1 Cauchy-Schwarz type inequalities......Page 354 12.2 Positive (semi)definite matrix inequalities......Page 357 12.3 Inequalities derived from the Schur complement......Page 373 12.4 Inequalities concerning eigenvalues......Page 375 Notes......Page 382 13 Matrix calculus......Page 383 13.1 Basic properties of differentials......Page 387 13.2 Scalar functions......Page 388 13.3 Vector functions......Page 392 13.4 Matrix functions......Page 393 13.5 The inverse......Page 396 13.6 Exponential and logarithm......Page 400 13.7 The determinant......Page 401 13.8 Jacobians......Page 405 13.9 Sensitivity analysis in regression models......Page 407 13.10 The Hessian matrix......Page 410 13.11 Least squares and best linear unbiased estimation......Page 414 13.12 Maximum likelihood estimation......Page 419 13.13 Inequalities and equalities......Page 423 Notes......Page 427 A.1 Some methods of indirect proof......Page 429 A.2 Primer on complex numbers and polynomials......Page 430 A.3 Series expansions......Page 433 A.3.1 Sequences and limits......Page 434 A.3.2 Convergence of series......Page 435 A.3.3 Special series......Page 436 A.3.4 Expansions of functions......Page 439 A.3.5 Multiple series, products, and their relation......Page 440 A.4.1 Linear difference equations......Page 441 A.4.3 Constrained optimization......Page 442 Notes......Page 446 B.1 Vectors and matrices......Page 447 B.2 Mathematical symbols, functions, and operators......Page 450 Bibliography......Page 455 Index......Page 458 "Matrix Algebra is the first volume of the Econometric Exercises series. It contains exercises relating to course material in matrix algebra that students are expected to know while enrolled in an (advanced) undergraduate or a postgraduate course in econometrics or statistics. The book contains a comprehensive collection of exercises, all with full answers. But the book is not just a collection of exercises; in fact, it is a textbook, though one that is organized in a completely different manner from the usual textbook. The volume can be used either as a self-contained course in matrix algebra or as a supplementary text."-- Jaquette "Matrix Algebra is the first volume of the Econometric Exercises series. It contains exercises relating to course material in matrix algebra that students are expected to know while enrolled in an (advanced) undergraduate or a postgraduate course in econometrics or statistics. The book contains a comprehensive collection of exercises, all with full answers. But the book is not just a collection of exercises; in fact, it is a textbook, though one that is organized in a completely different manner from the usual textbook. The volume can be used either as a self-contained course in matrix algebra or as a supplementary text."--Jacket