[wiley series in probability and statistics] methods and applications of linear models (regression and the analysis of variance) appendix a mathematics精品.pdfVIP

Methods and Applications ofLinear Models: Regression and the Analysis of Variance,2nd Edition. Ronald R. Hocking Copyright 0 2003 John Wiley Sons, Inc. ISBN: 0-471-23222-X Appendix A: Mathematics This appendix summarizes some of the results from matrix algebra and multivariate calculus utilized in the theoretical discussions in this book. It is assumed that the reader has had some exposure to this material, hence the results are presented primarily for reference. Most of the results on matrices are found in Graybill (1969) or Searle (1982). The results on Kronecker products are found in Neudecker (1969), Khuri (1983), and Smith and Hocking (1978). The material on optimization is found in most books on advanced calculus, or see Hadley (1 964). Additional detail is given for results not generally found in texts. A.1 MATRIX ALGEBRA A.1.1 Notation Matrices will be denoted by uppercase, boldfaced letters. To emphasize the elements of the rn x n matrix A with elements aij,i = 1, -..,rn, j = 1, .-.,n,we write A = (aii). The transpose of a matrix is the n x m matrix whose jth row is the jth column of A and is indicated by AT = (aji). The identity matrix is denoted by Z (or I, to emphasize its dimension). The inverse of a non-singular matrix is indicated by A- and has the property that A K 1 =A-A = I. (Section A.I.10 describes methods for obtaining the inverse matrix.) The product of two matrices, C =AB, where A is m x p and B is p x n, has elements D k=l The n x n matrix of ones is deno