20/11 DESIGN FOR A BRAIN 



where c is a constant, showing that under these conditions Xi 

 will converge to — c/au if an be negative, and will diverge without 

 limit if an be positive. 



If the diagonal terms an are much larger in absolute magnitude 

 than the others, the roots tend to the values of an. It follows 

 that if the diagonal terms take extreme values they determine 

 the stability. 



Example 3 : If the terms aij in the first n — 1 rows (or columns) 

 are given, the remaining n terms can be adjusted to make the 

 latent roots take any assigned values. 



Example 4 : The matrix of the homeostat equations of S, 19/11 

 is 



1 



a^h a 12 h a 13 h a li h 

 a 9 ,h a 99 h a 9 Ji a 9 Ji 



24' 



-J 



h a»Ji a^Ji anji 



33' 



-J 



_a il h a i9 h a i3 h a^Ji 



-1 



If j = o, the system must be unstable (by Example 1 above). 

 If the matrix has latent roots fi v . . . , /u 8 , and if A l5 . . . , A 4 

 are the latent roots of the matrix [a%jh] 9 and if j ^ 0, then the 

 A's and ^'s are related by X p = jbt 2 q -f- jju q . As ; — > oo the 8-variable 

 and the 4-variable systems are stable or unstable together. 



Example 5 : In a stable system, fixing a variable may make 

 the system of the remainder unstable. For instance, the system 

 with matrix 



6 5 - 10" 



- 4 - 3 - 1 

 4 2 - 6 . 



is stable. But if the third variable is fixed, the system of the 

 first two variables has matrix 



L-4 -3J 



and is unstable. 



Example 6 : Making one variable more stable intrinsically 



220 



