Dominant eigenvector displayed as third (Marco Visser)
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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