DESIGN FOR A BRAIN 



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Example 5: Making one variable more stable intrinsically 

 (Example 1 of this section) may make the whole unstable. For 

 instance, the system with matrix 



— 4 —31 

 3 2 J 



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



unstable when a n becomes more negative than 

 Example 6: In the n x n matrix 



a ! b 



*h 



in partitioned form, let the order of [a] be k x k. If the k diagonal 

 elements a u become much larger in absolute value than the rest, 

 the latent roots of the matrix tend to the k values a u and the 

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



1 — 3^ 



1 2 



has latent roots + 1-5 ± l-658t, 

 - 100 — 1 



and the matrix 

 2 0" 



2—100—1 2 



— 3 1—3 



2-1 1 2 



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



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 7: The holistic nature of stability is well shown by 

 the system with matrix 



— 3 — 2 2^ 



- 5 



5 6 



2 — 4 



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

 yet the whole is unstable. 



The probability of stability 



20/10. The probability that a system should be stable can be 

 made precise only after the system has been defined, ' stability ' 



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