Matrix Algebra From a Statistician's PerspectiveSpringer 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. |
Inhalt
Matrices | 1 |
Trace of a Square Matrix | 5 |
Inverse Matrices | 41 |
2 | 50 |
2 | 103 |
Idempotent Matrices | 133 |
10 | 139 |
Projections and Projection Matrices | 161 |
Exercises | 327 |
Kronecker Products and the Vec and Vech Operators 337 | 336 |
23 | 342 |
25 | 350 |
2 | 365 |
Generalized Inverses | 379 |
4 | 388 |
Their Sum | 396 |
1 | 166 |
3 | 177 |
Determinants | 179 |
3 | 187 |
6 | 195 |
4 | 205 |
Submatrices and Partitioned Matrices | 209 |
13 | 243 |
3 | 277 |
Matrix Differentiation | 289 |
Rank of a Matrix | 419 |
1 | 423 |
5 | 436 |
Minimization of a SecondDegree Polynomial in n Variables | 459 |
Minimization Problem | 468 |
Problem | 491 |
Linear System | 521 |
Linear Transformations | 589 |
References | 621 |
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according to Lemma according to Theorem column spaces continuously differentiable defined diagonal elements diagonal matrix dimensions equivalently essentially disjoint exists following corollary following lemma following theorem follows from Theorem formula function hence idempotent implying in light inner product interior point jth column Kronecker product LDU decomposition Let A represent light of Corollary light of Lemma light of Theorem linear space linear system AX linearly independent m n matrix n-dimensional column vector nonnegative definite matrix nonnull nonsingular matrix partitioned matrix permutation matrix positive definite matrix positive integers positive semidefinite principal submatrix projection matrix Proof PX;W q matrix R C STU rank(A represent an n n respect scalar Schur complement Section Show solution to AX submatrix Suppose symmetric matrix symmetric nonnegative definite symmetric positive definite U0DU usual inner product vech x'Ax