Matrix Algebra From a Statistician's PerspectiveSpringer Science & Business Media, 18.04.2006 - 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
| 1 | |
| 13 | |
3 Linear Dependence and Independence | 23 |
4 Linear Spaces Row and Column Spaces | 27 |
5 Trace of a Square Matrix | 49 |
6 Geometrical Considerations | 54 |
7 Linear Systems Consistency and Compatibility | 71 |
8 Inverse Matrices | 79 |
14 Linear Bilinear and Quadratic Forms | 209 |
15 Matrix Differentiation | 289 |
16 Kronecker Products and the Vec and Vech Operators | 336 |
17 Intersections and Sums of Subspaces | 379 |
18 Sums and Differences of Matrices | 419 |
19 Minimization of a SecondDegree Polynomial in n Variables Subject to Linear Constraints
| 459 |
20 The MoorePenrose Inverse | 496 |
21 Eigenvalues and Eigenvectors | 521 |
9 Generalized Inverses | 106 |
10 Idempotent Matrices | 133 |
11 Linear Systems Solutions | 139 |
12 Projections and Projection Matrices | 161 |
13 Determinants | 179 |
22 Linear Transformations | 589 |
| 621 | |
| 625 | |
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 à according to Lemma according to Theorem continuously differentiable defined diagonal elements diagonal matrix eigenvalue eigenvectors equivalently essentially disjoint exists following lemma following theorem follows from Theorem formula function hence idempotent ijth element implying in light inner product interior point inverse LDU decomposition Let A represent light of Corollary light of Lemma light of result light of Theorem linear space linear system linear system AX linear transformation linearly independent m n matrix matrix representation Moreover n n symmetric nonnegative n x n n-dimensional column vector nonnegative definite matrix nonnull nonsingular matrix orthogonal matrix orthonormal partitioned matrix permutation polynomial positive definite matrix positive semidefinite projection matrix Proof rank(A represent an n n respect scalar Schur complement Section Show subspace Suppose symmetric matrix symmetric nonnegative definite symmetric positive definite U0DU usual inner product x0Ax