Page 316 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

A scalar is a matrix with one row and one column.

A.2 Special Matrices

A.2.1 Square matrix
A matrix with an equal number of rows and columns is referred to as a square
matrix. Shown below is a square matrix G of order 3 × 3.
⎡ 21 6⎤
⎢ ⎥
G = 42 7 ⎥
⎢
⎢ ⎣ 04 8⎥ ⎦
The ij elements in a square matrix with i equal to j are called the diagonal elements.
Other elements of the square matrix are called off-diagonal or non-diagonal elements.
Thus the diagonal elements in the G matrix above are 2, 2 and 8.

A.2.2 Diagonal matrix

A square matrix having zero for all of its off-diagonal elements is referred to as a
diagonal matrix. For example, a diagonal matrix B can be shown as below:
⎡ 3 0 0⎤
⎢
B = 04 0 ⎥ ⎥
⎢
⎢ ⎣ 00 18⎥ ⎦

When all the diagonal elements of a diagonal matrix are one, it is referred to as an
identity matrix. Given below is an identity matrix, I:
⎡ 1 000⎤
⎢ 01 0 0 ⎥
I = ⎢ ⎥
⎢ 00 1 0⎥
⎢ ⎥
⎣ 000 1 ⎦

A.2.3 Triangular matrix

A square matrix with all elements above the diagonal being zero is called a lower
triangular matrix. When all the elements below the diagonal are zeros, it is referred
to as an upper triangular matrix. For instance, the matrices D, a lower triangular
matrix and E, an upper triangular matrix, can be illustrated as:

é 40 0ù é 39 1ù
ê
ú
D = ê ê 13 0 ; E = 048 ú ú
ú
ê
ë - ê 27 9ú û ê ë 00 6ú û
300 Appendix A