Theory Relevant to Modeling

Matrices are central to this book. In order to completely understand much of the material contained herein, a basic understanding of vectors, matrices, and linear algebra is required. There are a number of introductory texts on this material—every undergraduate school in the world probably offers a class in linear 

if X were the residuals from a linear model then XTX could be used to compute the residual sum of squares. A matrix is a group of columns all having the same number of rows. The number of rows in a matrix does not have to equal the number of columns, e.g., the matrix below has 4 rows and 5 columns 

The size of a matrix is the number of rows and columns hence, X in Eq. (A.4) is 4 x 5 (4 rows by 5 columns). The rank of a matrix is the number of linearly independent rows or columns (since the row and column rank of a matrix are the same). A matrix of size n x p that has rank less than p cannot be inverted (matrix inversion is discussed later). Similarly, a matrix of size n x n cannot be inverted if its rank is less than n. 

The values of a vector or matrix are typically enclosed by brackets, parentheses, or braces. Elements within a vector are identified by the matrix name followed by the row or column number in subscript. For example, X2 denotes the number 2 in Eq. (A.l). Identification of matrix elements are based on subscripts but, in this case, two subscripted integers are used the first integer in the subscript denotes the row while the second integer denotes the column. So, for example, Xu in Eq. (A.4) refers to the number T, X32 refers to the number T2, etc. To refer to an entire row, the V notation is often used. To refer to the entire second row in X, then X2 is used. To refer to the entire third column of X then X.3 is used. 

The transpose of a matrix, denoted with a superscript T, basically turns a matrix on its side such that the first row of the old matrix becomes the first column of the transposed matrix, e.g., in Eq. (A.4) 

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