STABILITY 



20/12 



(Example 2 of this section) may make the whole unstable. For 

 instance, the system with matrix 



is stable. But if a lx becomes more negative, the system becomes 

 unstable when a xl becomes more negative than — 4 J. 

 Example 7 : In the n x n matrix 



a 



c i d 



in partitioned form, [a] is of order k X k. If the k diagonal 

 elements an become much larger in absolute value than the rest, 

 the latent roots of the matrix tend to the k values an and the 

 n — k latent roots of [d]. Thus the matrix, corresponding to [d], 



1 

 .1 

 has latent roots -- 1-5 i l-658«, and the matrix 



— 1 2 0" 

 100-1 2 



— 3 1—3 



— 1 1 2. 



has latent roots -- 101-39, — 98-62, and + 1-506 ± 1-720*. 



Corollary : If system [d] is unstable but the whole 4-variable 

 system is stable, then making x x and x 2 more stable intrinsically 

 will eventually make the whole unstable. 



Example 8 : The holistic nature of stability is well shown by 

 the system with matrix 



— 3 — 2 2' 

 -6-5 6 



— 5 2 — 4_ 

 in which each variable individually, and every pair, is stable ; 

 yet the whole is unstable. 



100 



— 2 







2 



The probability of stability 



20/12. The probability that a system should be stable can be 

 made precise by the point of view of S. 14/16. We consider 



221 



