10 GENETIC STOCKS AND BREEDING METHODS 



The problem is to find A such that 



AGA" 1 = A 

 or 



AG = AA. 



AG is an s x s matrix, row i of which is a row vector which can be written afi, where a { 

 is row i of matrix A. AA is also anui matrix, row i of which can be written a^kj, 

 where X t is the non-zero element in row i of A, and I is an s x s identity matrix, with 

 diagonal elements 1 and all elements off the diagonal equal to zero. Then to satisfy (5), 



fl t G = fliXjI, 

 a ( G — fliXjI = 

 a,(G - X,I) = 0. (7) 



This defines a set of linear equations in the elements of a u the solutions of which give 



the elements of row i of A. If the values of X t from the matrix A are substituted into 



(7), the A matrix required to transform P to V and V n to P n is obtained. 



To obtain this solution, we must first have the A matrix. Equation (7) can be 



solved if and only if 



|G - X 4 I| = 0. (8) 



If X is subtracted from each of the diagonal elements of G and the determinant of 

 the resulting matrix set equal to zero, s solutions will be obtained for X, which constitute 

 the X b i = 1 , . . . , s, the diagonal elements of the matrix A. 



In each of the systems we shall discuss, one of the roots of equation (8) will be 1 . 

 Since the roots in A can be arranged in any order, we will always designate this root as 

 X s . The largest numerically of the remaining roots X l5 . . ., X s _! we will designate X x . 

 X x is often called the characteristic root of matrix G and has certain properties which 

 make it of special value to us. 



Since X 1 is the largest of the roots X l5 . . ., X s _ l5 all of which are less than 1, as the 

 matrix A is raised to higher powers, all of these elements become smaller (except X s , 

 which remains 1 for any power). Since X x is the largest of these, it will retain signifi- 

 cance even when the rest of the roots have become negligible. Thus, X x becomes an 

 approximate measure of the rate of change of frequency of the various mating types. 



We may also wish to know how many generations are required to exceed a given 

 percentage of incross matings. This will occur (ignoring sampling variations) when 



Pn > «, (9) 



if/> n is the proportion of incrosses and a is the desired proportion of such matings. 



If we assume that the initial generation includes only crosses which maximize 

 the number of generations required to reach a given percentage of incross matings, 



v = ap =: /r 

 1, 



