20/6 



DESIGN FOR A BRAIN 



20/6. Each coefficient rm is the sum of all i-vowed principal 

 (co-axial) minors of A, multiplied by (— 1)*. Thus, 



m l = — («11 + «22 + • • • + a nn) \ Wl„ = (— l) n | A |. 



Example : The linear system 



dxjdt = — 5x ± + 4a? 2 — 6^3! 



7a; 1 



6x 2 + 8x 3 

 4# 3 , 



dxjdt = 



dx 3 /dt = — 2x x + 4^2 



has the characteristic equation 



A 3 + 15A 2 + 21 + 8 = 



20/7. Of this equation the roots X lt . . . , A B are the latent 



roots of ^4. The integral of the canonical equations gives each 

 X{ as a linear function of the exponentials eV, . . . , eV. For 

 the sum to be convergent, no real part of A ls . . . , A n must be 

 positive, and this criterion provides a test for the stability of 

 the system. 



Example : The equation A 3 + 15A 2 -f 2 A + 8 = has roots 

 — 14-902 and — 0-049 ± 0-729 V^^T, so the system of the 

 previous section is stable. 



20/8. A test which avoids finding the latent roots is Hurwitz' : 

 a necessary and sufficient condition that the linear system is 

 stable is that the series of determinants 



etc. 



Example : The system with characteristic equation 

 P + 15A 2 + 2A + 8 = 

 yields the series 



+ 15, 



15 



8 



15 



8 

 





 15 



8 



These have the values + 15, + 22, and + 176. 

 is stable, agreeing with the previous test. 



218 



So the system 



