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

**Follow-Ups**:**Re: Re: smooth eigenvalues and eigenvectors as a function of frequency***From:*Maria Cristina Dias Tavares <cristina@dsce.fee.unicamp.br>