Page 319 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

The direct product is also known as the Kronecker product and is of the order nt by
ms. For instance, assuming that:
é 10 2ù
é 10 5ù ê ú
G = ê ú and A = 01 4 ú
ê
ë 520 û ê ë 24 1ú û

the Kronecker product is:

⎡ 10 020 5 010⎤
⎢ ⎥
⎢ 01040 0 5 20 ⎥
⎢ 20 40 10 10 20 5⎥
G ⊗ A = ⎢ ⎥
⎢ 5 0 10 20 0 40 ⎥
⎢ 0 5 20 0 20 80 ⎥
⎢ ⎥
⎣ ⎢ 10 20 5 20 80 20⎥ ⎦
0

The Kronecker product is useful in multiple trait evaluations.

A.3.5 Matrix inversion

An inverse matrix is one which when multiplied by the original matrix gives an identity
−1
matrix as the product. The inverse of a matrix A is usually denoted as A and, from
−1
the above definition, A A = I, where I is an identity matrix. Only square matrices can
be inverted and for a diagonal matrix the inverse is calculated simply as the reciprocal
of the diagonal elements. For instance, the diagonal matrix B and its inverse are:
é 1 ù
é 30 0ù ê 3 0 0 ú
ê ú ê ú
-
ê
1
B = 04 0 ú and B = 0 1 0ú ú
ê
ê ë 00 18ú û ê ê 4 1 ú ú
ê ë 0 0 18 ú û
For a 2 × 2 matrix, the inverse is easy to calculate and is illustrated below. Let:
é a11 a12ù
A = ê ú
ë a 21 a 22û

First, calculate the determinant, which is the difference between the product of the
two diagonal elements and the two off-diagonal elements (a a − a a ). Second,
11 22 12 21
the inverse is obtained by reversing the diagonal elements, multiplying the off-diagonal
elements by −1 and dividing all elements by the determinant. Thus:

1 é a 22 -a ù
12
A -1 = ê ú
12 21 ë
aa - a a -a 21 a 11û
11 22
Appendix A 303