Page 321 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

A.3.7 Generalized inverses

While an inverse does not exist for a singular matrix, a generalized inverse can, how-
−
ever, be calculated. A generalized inverse for a matrix D is usually denoted as D and
satisfies the expression:
−
DD D = D
Generalized inverses are not unique and may be obtained in several ways. One of the
simplest ways to calculate a generalized inverse of a matrix, say D in Section A.3.6,
is to initially obtain a matrix B of full rank as a subset of D. Set all elements of D to
zero. Calculate the inverse of B and replace the elements of D with corresponding
−
elements of B and the result is D . For instance, for the matrix D above, the matrix B,
a full rank subset of D, is:
é 32ù é 3 - 2ù
B = ê ú and B -1 = ê ú
ë 43 û ë - 4 3 û
Replacing elements of D with the corresponding elements of B after all elements of D
−
have been set to zero gives D as:
⎡ 3 −2 0⎤
− ⎢ ⎥
D = ⎢ −4 3 0 ⎥
⎢ ⎣ 0 0 0⎥ ⎦

A.3.8 Eigenvalues and eigenvectors

Eigenvalues are also referred to as characteristic or latent roots and are useful in
simplifying multivariate evaluations when transforming data. The sum of the eigen-
values of a square matrix equals its trace (sum of the diagonal elements of a square
matrix) and their product equals its determinant (Searle, 1982). For symmetric matri-
ces, the rank equals the number of non-zero eigenvalues.
For a square matrix B, the eigenvalues are obtained by solving:
|B − dI| = 0
where the vertical lines denote finding the determinant.
With the condition specified in the above equation, B can be represented as:
BL = LD
B = LDL −1 (a.1)
where D is a diagonal matrix containing the eigenvalues of B, and L is a matrix of
corresponding eigenvectors. The eigenvector (k) is found by solving:

(B − d I)l = 0
k k
where d is the corresponding eigenvalue.
k
−1
For symmetric matrices L is orthogonal (that is, L = L′; LL′ = I = L′L); therefore,
given that B is symmetric, (Eqn a.1) can be expressed as:
B = LDL′
Usually, eigenvalues and eigenvectors are calculated by means of computer programs.

Appendix A 305