[R] Dominant eigenvector displayed as third (Marco Visser)

[RAVI VARADHAN]

Yes, Spencer, your observation is correct, because the characeristic equation det(A - \lambda*I) is a sixth degree polynomial: \lambda^6 - 5 = 0.  So the eigenvalues are the complex numbers (generally) that are located at equal angles on the circle of radius 5^(1/6), at angles 2*pi*k/6, where k runs from 0 to 5.  Thus, the roots are:

z_k = 5^(1/6) * exp(i * 2*pi*k/6), k= 0, 1, ..., 5.

where i = sqrt(-1).

Ravi.

----- Original Message -----
From: Spencer Graves <spencer.graves at pdf.com>
Date: Friday, June 29, 2007 6:51 pm
Subject: Re: [R] Dominant eigenvector displayed as third (Marco Visser)
To: Marco Visser <visser_md at yahoo.com>
Cc: r-help at stat.math.ethz.ch


>       There is no dominant eigenvalue:  The eigenvalues of that matrix 
> 
>  are the 6 different roots of 5.  All have modulus (or absolute value) 
> = 
>  1.307660.  When I raised them all to the 6th power, all 6 were 5+0i. 
> 
>  
>        Someone else can tell us why this is, but this should suffice 
> as 
>  an initial answer to your question. 
>  
>        Hope this helps. 
>        Spencer Graves
>  
>  Marco Visser wrote:
>  > Dear R users & Experts,
>  >
>  > This is just a curiousity, I was wondering why the dominant 
> eigenvetor and eigenvalue 
>  > of the following matrix is given as the third. I guess this could 
> complicate automatic selection 
>  > procedures. 
>  >
>  > 0    0    0    0    0    5
>  > 1    0    0    0    0    0
>  > 0    1    0    0    0    0
>  > 0    0    1    0    0    0
>  > 0    0    0    1    0    0
>  > 0    0    0    0    1    0
>  >
>  > Please copy & paste the following into R;
>  >
>  > a=c(0,0,0,0,0,5,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0)
>  > mat=matrix(a, ncol=6,byrow=T)
>  > eigen(mat)
>  >
>  > The matrix is a population matrix for a plant pathogen (Powell et 
> al 2005).
>  >
>  > Basically I would really like to know why this happens so I will 
> know if it can occur 
>  > again. 
>  >
>  > Thanks for any comments,
>  >
>  > Marco Visser
>  >
>  >
>  > Comment: In Matlab the the dominant eigenvetor and eigenvalue 
>  > of the described matrix are given as the sixth. Again no idea why.
>  >
>  > reference
>  >
>  > J. A. Powell, I. Slapnicar and W. van der Werf. Epidemic spread of 
> a lesion-forming 
>  > plant pathogen - analysis of a mechanistic model with infinite age 
> structure. (2005) 
>  > Linear Algebra and its Applications 298. p 117-140.  
>  >
>  >
>  >
>  >
>  >        
>  > 
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