Page 315 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Appendix A: Introduction to Matrix

Algebra

The basic elements of matrix algebra necessary to understand the principles involved
in the prediction of breeding values are briefly covered in this appendix. Little or no
previous knowledge of matrix algebra is assumed. For a detailed study of matrix
algebra, see Searle (1982).

A.1 Matrix: A Definition

A matrix is a rectangular array of numbers set in rows and columns. These elements
are called the elements of a matrix. The matrix B, for instance, consisting of two rows
and three columns, may represented as:
é b11 b12 b13ù
B = ê ú
ë b21 b22 b 23û
or:
2 ⎡ 45⎤
B = ⎢ ⎥
⎣ 68 9 ⎦
The element b is called the ij element of the matrix, the first subscript referring to
ij
the row the element is in and the second to the column. The order of a matrix is the
number of rows and columns. Thus a matrix of r rows and c columns has order r × c
(read as r by c). The matrix B above is of the order 2 × 3 and can be written as B .
2×3
A matrix consisting of a single row of elements is called a row vector. A row vec-
tor consisting of three elements may be represented as:
[ 4]
c = 2 6 -
Only one subscript is needed to specify the position of an element in a row vector.
Thus the ith element in the row vector c above refers to the element in the ith column.
For instance, c = −4.
3
Similarly, a matrix consisting of a single column is called a column vector. Again,
only one supscript is needed to specify the position of an element, which refers to the
row the element is in, since there is only one column. A column vector d with four
elements can be shown as below:
- é 20ù
ê 60 ú
d = ê ú
ê 8ú
ê ú
ë 2 û

© R.A. Mrode 2014. Linear Models for the Prediction of Animal Breeding Values, 299
3rd Edition (R.A. Mrode)