R help - May 2015 - order by which the eigenvalues are presented

Thank you, Bill, for your reply. However, I'm afraid I didn't explain
myself properly.

 

Imagine you have a 2x2 matrix

Then the eigenvalues lambda_1 and lambda_2 are analytically calculated from

 

lambda_1 = (-b+sqrt(delta))/2a

lambda_2 = (-b-sqrt(delta))/2a

 

where delta = b^2-4ac

 

If delta>0 then lambda_1 > lambda_2 always. Otherwise their Real parts are
equal.

 

If we have a 3x3 matrix the three eigenvalues will have very complicated
expressions:

 

lambda_1 = f_1

lambda_2 = f_2

lambda_3 = f_3

 

where f_1,f_2 and f_3 are functions of the elements of the matrix
a11,a12...,a33, which are sampled from a given distribution (e.g. normal(0,1)).

 

What I would like to know is from which expression (f_1,f_2 or f_3) comes the
largest Re part of the eigenvalues. For example, does it always come from f_1
independently of the sampled values of a11,a12...,a33?

 

Thank you



____________________________________________________________
Lu?s Borda de ?gua
REFER Biodiversity Chair
CIBIO - Research Center in Biodiversity and Genetic Resources
Campus Agr?rio de Vair?o
R. Padre Armando Quintas
4485-661 Vair?o, Portugal

IICT - Tropical Research Institute
Travessa Conde da Ribeira N. 9 R/C
1300-142 Lisboa, Portugal
Tel: +351 21 361 63 40 ext. 312


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On 06/05/2015 12:58 PM, Luis Borda de Agua wrote:
> Thank you, Bill, for your reply. However, I'm afraid I didn't
explain myself properly.
> 
>  
> 
> Imagine you have a 2x2 matrix
> 
> Then the eigenvalues lambda_1 and lambda_2 are analytically calculated from
> 
>  
> 
> lambda_1 = (-b+sqrt(delta))/2a
> 
> lambda_2 = (-b-sqrt(delta))/2a
> 
>  
> 
> where delta = b^2-4ac
> 
>  
> 
> If delta>0 then lambda_1 > lambda_2 always. Otherwise their Real
parts are equal.
The first condition is incomplete.  You need a>0 as well.
> 
>  
> 
> If we have a 3x3 matrix the three eigenvalues will have very complicated
expressions:
> 
>  
> 
> lambda_1 = f_1
> 
> lambda_2 = f_2
> 
> lambda_3 = f_3
> 
>  
> 
> where f_1,f_2 and f_3 are functions of the elements of the matrix
a11,a12...,a33, which are sampled from a given distribution (e.g. normal(0,1)).
> 
>  
> 
> What I would like to know is from which expression (f_1,f_2 or f_3) comes
the largest Re part of the eigenvalues. For example, does it always come from
f_1 independently of the sampled values of a11,a12...,a33?
I don't know the answer, but I would expect it to depend on the entries
in the matrix in quite a complicated way.

Duncan Murdoch