From (3.6), 2 3
 J =6 7: (3.12)
4 52
b 2 233dp dp3
In order to evaluate the eigenvalues of J, we need to obtain the following determinant
  
    
2 5 2 = 2 + ( +  + dp
3) + (b +  + dp3): (3.13) 
2 b 2dp  3
3 
2dp
3  3 
We will need to solve the following equation
2 + B + C = 0; (3.14)
where
2B =  +  + dp
3 (3.15)
=  + b (3.16)
5 2C = (b +  + dp
3) (3.17)3
 
= 2 (b): (3.18)3
The eigenvalues of J are real numbers because, the discriminant of the above quadratic is:
 

= ( + b)2 4 2 (b) (3.19)3
= ( + b)2 8+  (b) (3.20)3
8= ( + b)2 8+  b (3.21)
3 3
= ( b)2 8+ 4b +  8b (3.22)3 3
= ( 4b)2 8+  + b > 0 (3.23)3 3

> 0 (3.24)
11
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