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MathGroup Archive 2005

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Re: smooth eigenvalues and eigenvectors as a function of frequency

  • To: mathgroup at smc.vnet.net
  • Subject: [mg60321] Re: smooth eigenvalues and eigenvectors as a function of frequency
  • From: "Eckhard Hennig" <aidev at n-o-s-p-a-m.kaninkolo.de>
  • Date: Tue, 13 Sep 2005 06:06:55 -0400 (EDT)
  • References: <dg05rf$a1u$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

"Antonio Carlos Siqueira" <acsl at dee.ufrj.br> schrieb im Newsbeitrag 
news:dg05rf$a1u$1 at smc.vnet.net...
> Dear MathGroup
>
> I am posting this message hoping that someone may have a better idea
> than me and point me in some direction to the solution. I have to fit a
> complex based function using a state-space approach and thus smooth
> eigenvectors and eigenfunctions are needed. Using Eigensystem I
> experienced some eigenvector/eigenvalues switchovers (from one
> frequency step to the next).
> I was wondering whether some sort of MapIndexed or some Sort can do the
> trick to switch the eigenvector back. I know that if I can track the
> direction of the eigenvalue.eigenvector dot product I might probably
> identify the switchover.

Hi Antonio,

without having looked into all the details of your code (it's not complete 
anyway) I believe that your problem is one of identifying corresponding 
eigenvectors of two closely related eigenvalue problems: Let A(p) denote a 
square matrix whose entries are functions of a parameter p, and let deltap 
denote a (small) perturbation of p. Let the two eigenvalue problems be 
formulated as

(1)     (A(p) - lambda.I) x = 0

and

(2)     (A(p+deltap) - lambda.I) x = 0

Now assume that (lambda1_1, x1_1) is an eigenvalue/eigenvector pair of (1) 
and that you are interested in finding the corresponding pair (lambda2_1, 
x2_1) in the set of solutions {(lambda2_k, x2_k), k=1..n} of the perturbed 
equation (2). This can be achieved with the help of the modal assurance 
criterion (MAC), defined as

  MAC(x1, x2) = abs(x1'.x2)^2/((x1'.x1)*(x2'.x2))

where x' denotes the complex conjugate transpose of x.

The MAC ranges from 0 to 1. The eigenpair of (2) corresponding to 
(lambda1_1, x1_1) is the one for which

  MAC(x1_1, x2_k)

is closest to 1.

HTH,

Eckhard

-- 
Dr.-Ing. Eckhard Hennig
www.kaninkolo.de/ai
aidev \at kaninkolo \dot de



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