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Re: different eigenvectors

  • To: mathgroup at smc.vnet.net
  • Subject: [mg83903] Re: different eigenvectors
  • From: rip pelletier <bitbucket at comcast.net>
  • Date: Tue, 4 Dec 2007 04:21:11 -0500 (EST)
  • References: <fj0rjv$per$1@smc.vnet.net>

In article <fj0rjv$per$1 at smc.vnet.net>,
 vicky Al Aisa <vickyisai at gmail.com> wrote:

> I am using Eigenvectors function to calculate eigenvectors in
> mathematica 5.2.
> while comparing the results with the eigenvectors (of same matrix)
> computed in another system, i found that both the results are bit different
> with some values having opposite sign and coloumns rearranged.
> 
hi vicky,

both behaviors are to be expected. that said, there is no harm in 
verifying that each eigenvalue-eigenvector pair really works; in fact, 
you should do that for a while with any program new to you. (use 
eigensystem in mathematica to get both eigenvectors & eigenvalues.)

first, eigenvectors are not unique: if v is an eigenvector, then so is 
a*v for any scalar a ? 0. and even if v is of unit length, so is -v. 
(we're really finding an entire subspace, not just a single vector.)

second, the order of the eigenvalues is not unique: if the columns are 
rearranged, the eigenvalues need to be rearranged, but so long as they 
correspond, things are fine. (geometrically, we're just changing the 
order of the basis vectors.)

my favorite intro linear algebra book is "linear algebra & its 
applications", by gilbert strang. a 4th ed. just came out, so you might 
find a 3rd ed cheaper if that matters.

vale,
   rip

-- 
NB eddress is r i p 1 AT c o m c a s t DOT n e t


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