20/9 



STABILITY 



where c is a constant, showing that under these conditions x t 

 will converge to — c/a H if a u be negative, and will diverge without 

 limit if a u be positive. 



If the diagonal terms a u are much larger in absolute magnitude 

 than the others, the latent roots tend to the values of a u . It 

 follows that if the diagonal terms take extreme values they 

 determine the stability. 



Example 2: If the terms a u 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 3: The matrix of the Homeostat equations of S. 19/11 

 is 



fliJi a-iJn a,Ji 



[ n 



a 2 Ji 

 a zl h 

 a A Ji 



12' 



a 22 h 

 a» 9 h 



13' 



a 23 h 

 a zz h 

 a i3 h 



a 



a 24 

 a 34 



a 



J. 



If j = 0, the system must be unstable, for the eight latent roots 

 are the four latent roots of [a tJ ], each taken with both positive and 

 negative signs. If the matrix has latent roots p l9 . . . , // 8 , and 

 if A 1? . . . , A 4 are the latent roots of the matrix [a^h], and if 

 j ^ 0, then the A's and ^'s are related by X v = ju^ + j/^a- As 

 j — > ± °o the 8-variable and the 4-variable systems are stable 

 or unstable together. 



Example 4: In a stable system, fixing a variable may make 

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

 with matrix 



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

 first two variables has matrix 



6 5" 



4 - 3 



and is unstable. 



257 



[_: _g 



