Page 317 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

The transpose (see A.3.1) of an upper triangular matrix is a lower triangular matrix
and vice versa.

A.2.4 Symmetric matrix

A symmetric matrix is a square matrix with the elements above the diagonal equal to
the corresponding elements below the diagonal, i.e. element ij is equal to element ji.
The matrix A below is an example of a symmetric matrix:
é 2 - 4 0ù
ê
A = - 4 6 3 ú ú
ê
ê ë 0 3 7ú û

A.3 Basic Matrix Operations

A.3.1 Transpose of a matrix

The transpose of a matrix A is usually written as A′ or A and is the matrix whose ji
T
elements are the ij elements of the original matrix, i.e. a′ = a . In other words, the
ji ij
columns of A′ are the rows of A and the rows of A′ the columns of A. For instance,
the matrix A and its transpose A′ are illustrated below:
⎡ 32⎤
⎢ ⎥ ⎡ 31 4⎤
A = 11 ; A′ = ⎢ ⎥
⎢
⎥
⎢ ⎣ 40⎥ ⎦ ⎣ 21 0 ⎦
Note that A is not equal to A′ but the transpose of a symmetric matrix is equal to
the symmetric matrix. Also (AB)′ = B′A′, where AB refers to the product (see A.3.3)
of A and B.

A.3.2 Matrix addition and subtraction

Two matrices can be added together only if they have the same number of rows and
columns, i.e. they are of the same order and they are said to be conformable for addi-
tion. Given that W is the sum of the matrices X and Y, then w = x + y . For example,
ij ij ij
if X and Y, both of order 2 × 2, are as illustrated below:
é 40 10ù - é 220ù
X = ê ú ; Y = ê ú
ë 39 - 25 û ë 4 40 û
Then the matrix W, the sum of X and Y, is:

-
é 40 + ( 2) 10 + 20ù é 38 30ù
W = ê ú ê ú
=
ë 39 + 4 - 25 + 40 û ë 43 15 û
Appendix A 301