SYSTEMS OF MATING 9 



This means that if the G matrix could be raised to the power n, the mating-type fre- 

 quencies could be obtained immediately in generation n. Raising G to the power n is, 

 however, a difficult task. To avoid this difficulty the method of Kempthorne can be 

 used. 700 



A diagonal matrix is one in which some of the elements of the leading diagonal 

 have values other than 0, while all elements off this diagonal are zero. If the matrix 

 A is an s x s diagonal matrix, 



A = / X x 



X 2 



A raised to the power n is 



A n = /Xj" 



X,* 



X s " 



Thus, if G were a diagonal matrix, we could easily obtain G n . G will not, however, 

 be diagonal for any system of inbreeding; but we can use a transformation which will 

 give us a diagonal matrix to work with. Let us define an s x s matrix, A, such that 



V fc = AP fc (4) 



for any generation k, V fc being an s x 1 vector. Then from equation (2), 



V fc = AGP fc _ l5 



and since 



it follows that 



V fc = AGA- 1 ^-!. (5) 



But an s x s matrix A exists such that AGA~ 1 is an s x s diagonal matrix. 

 Therefore, 



V fc = AV fc _ l5 



and, since this has the same form as (2), 



V n = A"V . (6) 



If, then, we can obtain A and A, we can transform P to give us V , raise A to the power 

 n, calculate V n , and transform to P n . 



