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 <[EMAIL PROTECTED]> Date: Friday, June 29, 2007 6:51 pm Subject: Re: [R] Dominant eigenvector displayed as third (Marco Visser) To: Marco Visser <[EMAIL PROTECTED]> Cc: r-help@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. > > > > > > > > > > > > > ____________________________________________________________________________________Ready > > for the edge of your seat? > > Check out tonight's top picks on Yahoo! TV. > > > > [[alternative HTML version deleted]] > > > > ______________________________________________ > > R-help@stat.math.ethz.ch mailing list > > > > PLEASE do read the posting guide > > and provide commented, minimal, self-contained, reproducible code. > > > > ______________________________________________ > R-help@stat.math.ethz.ch mailing list > > PLEASE do read the posting guide > and provide commented, minimal, self-contained, reproducible code. ______________________________________________ R-help@stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.